• diferenţa Show Notations de bg tr

    Fie AA si BB doua mulţimi, atunci diferenţa A\Bhttp://mathhub.info/smglom/sets/setdiff.omdoc?setdiff?set-differenceAB dintre AA şi BB este (bsetst[x]x)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstxx.


  • disjuncte de en tr

    O familie de mulţimi e considerată familie de mulţimi disjuncte , dacă oricare două mulţimi sunt disjuncte.

    Două mulţimi AA şi BB sunt considerate disjuncte, dacă (AB)=http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/sets/intersection.omdoc?intersection?intersectionABhttp://mathhub.info/smglom/sets/emptyset.omdoc?emptyset?eset.


  • egale Show Notations tr de ru

    Două mulţimi AA şi BB sunt egale (scris ABhttp://mathhub.info/smglom/sets/set.omdoc?set?setequalAB), dacă au aceleaşi elemente.


  • familie de mulţimi disjuncte , de en tr

    O familie de mulţimi e considerată familie de mulţimi disjuncte , dacă oricare două mulţimi sunt disjuncte.


  • familie de mulţimi disjuncte , de en tr

    O familie de mulţimi e considerată familie de mulţimi disjuncte , dacă oricare două mulţimi sunt disjuncte.


  • intersecţia Show Notations tr de en

    Fie II o mulţime şi SiOMFOREIGNscala.xml.Node$@6040b6ef o familie de mulţimi indexată de II, atunci intersecţia iISihttp://mathhub.info/smglom/sets/intersection.omdoc?intersection?mintersectCollectionIiOMFOREIGNscala.xml.Node$@6040b6ef peste II este (bsetst[x]x)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstxx.


  • intersecţia Show Notations tr de en

    Fie AA si BB doua mulţimi, atunci intersecţia ABhttp://mathhub.info/smglom/sets/intersection.omdoc?intersection?intersectionAB lui AA şi BB este (bsetst[x]x)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstxx.


  • mulţimea tuturor submulţimilor Show Notations de tr bg

    Fie AA o mulţime, atunci mulţimea tuturor submulţimilor 𝒫(A)http://mathhub.info/smglom/sets/powerset.omdoc?powerset?powersetA lui AA este (bsetst[S]S)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstSS.


  • Mulţimea vidă Show Notations en tr de

    Mulţimea vidă http://mathhub.info/smglom/sets/emptyset.omdoc?emptyset?eset (uneori scrisă http://mathhub.info/smglom/sets/emptyset.omdoc?emptyset?eset) este mulţimea fără nici un element.


  • pereche Show Notations tr en de

    Fie AA şi BB doua mulţimi, atunci mulţimea de perechi A×Bhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAB de AA şi BB este definită ca (bsetst[ab]a,b)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstabhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairab, numim (a,b)(A×B)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairabhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAB o pereche.


  • perechi Show Notations tr en de

    Fie AA şi BB doua mulţimi, atunci mulţimea de perechi A×Bhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAB de AA şi BB este definită ca (bsetst[ab]a,b)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstabhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairab, numim (a,b)(A×B)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairabhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAB o pereche.


  • reuniunea Show Notations de tr ru

    Fie II o mulţime şi (bsetst[i]Si)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstiOMFOREIGNscala.xml.Node$@6040b6ef o familie de mulţimi, atunci reuniunea iISihttp://mathhub.info/smglom/sets/union.omdoc?union?munionCollectionIiOMFOREIGNscala.xml.Node$@6040b6ef peste SS este (bsetst[x]x)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstxx.


  • reuniunea Show Notations de tr ru

    Fie AA şi BB două mulţimi, atunci reuniunea ABhttp://mathhub.info/smglom/sets/union.omdoc?union?unionAB lui AA şi BB este definită ca (bsetst[x]x)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstxx


  • submulţime Show Notations de tr en

    O mulţime AA este o submulţime a unei mulţimi BB (scris ABhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?proper-subsetAB), dacă toate xAhttp://mathhub.info/smglom/sets/set.omdoc?set?insetxA sunt elemente ale lui BB.


  • submulţime proprie Show Notations de tr en

    O mulţime AA este o submulţime proprie a unei mulţimi BB (scris ABhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetAB), dacă ABhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?proper-subsetAB şi ABhttp://mathhub.info/smglom/sets/set.omdoc?set?nsetequalAB.


  • super-mulţime Show Notations de tr en

    O mulţime AA este o super-mulţime a unei mulţimi BB (scris ABhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?proper-supersetAB), dacă BAhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?proper-subsetBA.


  • super-mulţime proprie Show Notations tr en

    O mulţime AA este o super-mulţime proprie a unei mulţimi BB (scris ABhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?supersetAB), dacă BAhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetBA.


  • Show Notations de

    The number ee, sometimes called Euler’s constant (also known as Napier’s constant) is an important mathematical constant.

    ehttp://mathhub.info/smglom/numberfields/eulersnumber.omdoc?eulersnumber?eulersnumber
    e=2.718281828459045...http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?startswithhttp://mathhub.info/smglom/numberfields/eulersnumber.omdoc?eulersnumber?eulersnumber

  • times differentiable de

    We call ff nn times differentiable at aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM, iff dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa exists as a limit. Analogously, ff is nn times differentiable on MM, iff dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx exists and is total on MM.

    ff is called infinitely differentiable or smooth at aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM or on MM, iff dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa and dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx exist for all nhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number respectively.


  • -closure de

    Let pp be a properties and R(A×B)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetRhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAB a relation, then we call the smallest (in terms of the subsethttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subset) relation RRhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?supersetOMFOREIGNscala.xml.Node$@6040b6efR that has property pp the pp-closure of RR.


  • -dim Cartesian space Show Notations de

    Let AA be a set, then the nn-dim Cartesian space Anhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?nCartSpaceAn over AA is (bsetst[an](a1,,an))http://mathhub.info/smglom/sets/set.omdoc?set?bsetstanhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlian. We call ((a1,,an))(An)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlianhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?nCartSpaceAn a vector.


  • -equal de

    We call a formula AA an alphabetic variant of BB (or http://mathhub.info/smglom/smglom/alpharenaming.omdoc?alpharenaming?alphaeqRel-equal; write (alphaeqA)Bhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/alpharenaming.omdoc?alpharenaming?alphaeqAB), iff BB can be obtained from AA by systematically renaming bound variables.


  • -fold Cartesian product Show Notations de

    Let AiOMFOREIGNscala.xml.Node$@6040b6ef be a collection of sets for 1inhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?betweeneein, then the nn-fold Cartesian product A1×...×Anhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?ncartliAn is (bsetst[an](a1,,an))http://mathhub.info/smglom/sets/set.omdoc?set?bsetstanhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlian, we call ((a1,,an))(A1×...×An)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlianhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?ncartliAn an nn-tuple.

    We call the function (projectioni):(A1×...×An)Ai;((a1,,an))aihttp://mathhub.info/smglom/sets/functions.omdoc?functions?funsuchthathttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?projectionihttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?ncartliAnOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlianOMFOREIGNscala.xml.Node$@6040b6ef the (ii th ) projection.


  • -ful number de

    An integer whose prime factors have exponents at least kk is called a kk-powerful number, kk-ful number, or kk-full number.


  • -full number de

    An integer whose prime factors have exponents at least kk is called a kk-powerful number, kk-ful number, or kk-full number.


  • -perfect de

    A number nn is called multiperfect or kk-perfect for a given natural numbers kk, if and only if the sum of all positive divisors of nn is equal to knhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationkn.


  • -powerful number de

    An integer whose prime factors have exponents at least kk is called a kk-powerful number, kk-ful number, or kk-full number.


  • -powersmooth de

    A natural numbers nn is called BB-powersmooth if all prime powers pkhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationpk dividing nn satisfy: (pk)<Bhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthanhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationpkB


  • -regular de

    If in a graph GG all the vertices have the same degree, say kk, then GG is called kk-regular, or simply regular.

    A 3-regular graph is called cubic graph.


  • -simplex de

    A kk-simplex is a kk-dimensional convex polytope which is the convex hull of k+1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionk affinely independent points in khttp://mathhub.info/smglom/linear-algebra/nspace.omdoc?nspace?Euclidean-spacek.


  • -tuple Show Notations de

    Let AiOMFOREIGNscala.xml.Node$@6040b6ef be a collection of sets for 1inhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?betweeneein, then the nn-fold Cartesian product A1×...×Anhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?ncartliAn is (bsetst[an](a1,,an))http://mathhub.info/smglom/sets/set.omdoc?set?bsetstanhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlian, we call ((a1,,an))(A1×...×An)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlianhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?ncartliAn an nn-tuple.

    We call the function (projectioni):(A1×...×An)Ai;((a1,,an))aihttp://mathhub.info/smglom/sets/functions.omdoc?functions?funsuchthathttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?projectionihttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?ncartliAnOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlianOMFOREIGNscala.xml.Node$@6040b6ef the (ii th ) projection.


  • -Ulam number de

    The u,vhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairuv-Ulam numbers Unhttp://mathhub.info/smglom/numbers/uvulamnumber.omdoc?uvulamnumber?uvulamnumbern form an integer sequence. The u,vhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairuv-Ulam sequence starts with (U1)=uhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/uvulamnumber.omdoc?uvulamnumber?uvulamnumberu and (U2)=vhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/uvulamnumber.omdoc?uvulamnumber?uvulamnumberv. Then for n>2http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethann, Unhttp://mathhub.info/smglom/numbers/uvulamnumber.omdoc?uvulamnumber?uvulamnumbern is defined to be the smallest integer that is the sum of two distinct earlier terms in exactly one way.


  • -Ulam sequence de

    The u,vhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairuv-Ulam numbers Unhttp://mathhub.info/smglom/numbers/uvulamnumber.omdoc?uvulamnumber?uvulamnumbern form an integer sequence. The u,vhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairuv-Ulam sequence starts with (U1)=uhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/uvulamnumber.omdoc?uvulamnumber?uvulamnumberu and (U2)=vhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/uvulamnumber.omdoc?uvulamnumber?uvulamnumberv. Then for n>2http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethann, Unhttp://mathhub.info/smglom/numbers/uvulamnumber.omdoc?uvulamnumber?uvulamnumbern is defined to be the smallest integer that is the sum of two distinct earlier terms in exactly one way.


  • 2-almost prime de

  • 2-almost prime de

  • 2-full numbers de

    A powerful number is a positive integer mm such that for every prime number pp dividing mm, p2http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationp also divides mm.

    A powerful number is the product of a square and a cube. Powerful numbers are also known as squareful numbers, square-full numbers, or 2-full numbers.


  • absolute value Show Notations de

    The absolute value |r|http://mathhub.info/smglom/numberfields/absolutevalue.omdoc?absolutevalue?absolute-valuer of a real number rr is defined as (defined-piecewise(
    rwenn(r0)
    )
    (
    (r)sonst
    )
    )
    http://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecerhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methanrhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionr
    .


  • absolutely abnormal de

    A number is absolutely non-normal or absolutely abnormal if it is not simply normal in any base.


  • absolutely non-normal de

    A number is absolutely non-normal or absolutely abnormal if it is not simply normal in any base.


  • abstract reduction system

    An abstract reduction system (or abstract rewriting system, or ARS) A,Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureAR consists of a set AA together with a relation R(A2)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetRhttp://mathhub.info/smglom/sets/pair.omdoc?pair?twodimA. The relation RR is written as (arsRconvR)http://mathhub.info/smglom/smglom/abstract-reduction-system.omdoc?abstract-reduction-system?arsRconvR or simply as arsconvhttp://mathhub.info/smglom/smglom/abstract-reduction-system.omdoc?abstract-reduction-system?arsconv.

    The transitive-reflexive closure of arsconvhttp://mathhub.info/smglom/smglom/abstract-reduction-system.omdoc?abstract-reduction-system?arsconv is denoted by arsconvtrhttp://mathhub.info/smglom/smglom/abstract-reduction-system.omdoc?abstract-reduction-system?arsconvtr. To distinguish the original relation arsconvhttp://mathhub.info/smglom/smglom/abstract-reduction-system.omdoc?abstract-reduction-system?arsconv it is sometimes written as arsconvhttp://mathhub.info/smglom/smglom/abstract-reduction-system.omdoc?abstract-reduction-system?arsconv.


  • abstract rewriting system

    An abstract reduction system (or abstract rewriting system, or ARS) A,Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureAR consists of a set AA together with a relation R(A2)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetRhttp://mathhub.info/smglom/sets/pair.omdoc?pair?twodimA. The relation RR is written as (arsRconvR)http://mathhub.info/smglom/smglom/abstract-reduction-system.omdoc?abstract-reduction-system?arsRconvR or simply as arsconvhttp://mathhub.info/smglom/smglom/abstract-reduction-system.omdoc?abstract-reduction-system?arsconv.

    The transitive-reflexive closure of arsconvhttp://mathhub.info/smglom/smglom/abstract-reduction-system.omdoc?abstract-reduction-system?arsconv is denoted by arsconvtrhttp://mathhub.info/smglom/smglom/abstract-reduction-system.omdoc?abstract-reduction-system?arsconvtr. To distinguish the original relation arsconvhttp://mathhub.info/smglom/smglom/abstract-reduction-system.omdoc?abstract-reduction-system?arsconv it is sometimes written as arsconvhttp://mathhub.info/smglom/smglom/abstract-reduction-system.omdoc?abstract-reduction-system?arsconv.


  • abundance de

    The abundance of a positive integer nn is the value (σ1(n))-(2n)http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/smglom/sumofdivisorsfunction.omdoc?sumofdivisorsfunction?sumdiv-functionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationn.


  • abundancy de

    The abundancy of a number nn is defined as the ratio (σ1(n))nhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/smglom/sumofdivisorsfunction.omdoc?sumofdivisorsfunction?sumdiv-functionnn, where σ1(n)http://mathhub.info/smglom/smglom/sumofdivisorsfunction.omdoc?sumofdivisorsfunction?sumdiv-functionn is the sum-of-divisors function.


  • Achilles number de

    An Achilles number is a natural numbers that is powerful but not a perfect power.


  • acyclic

    Given a directed graph G=(V,E)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalGhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE,

    • a path pp is called cyclic (or a cycle) iff (start(p))=(end(p))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pstartphttp://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pendp.

    • a cycle (v0,,vn)http://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlivn is called simple, iff vivjhttp://mathhub.info/smglom/mv/equal.omdoc?equal?notequalOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6ef for i,j1nhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?mbetweeneeijn with ijhttp://mathhub.info/smglom/mv/equal.omdoc?equal?notequalij.

    • GG is called acyclic (or a DAG (directed acyclic graph)) iff there is no cycle in GG.


  • additive persistence

    The number of times the digits must be summed to reach the digital sum is called a number’s additive persistence.


  • additive structure de

    A structure Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR is called a ring, if Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR is a Abelian group (called the additive structure), Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR is a monoid (called the multiplicative structure), and Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR is a ringoid.

    We call 0 the zero of the ring and 1 the one of the ring.


  • adjacent de

    Two distinct edges ee and ff are adjacent if they have an end in common.

    Two vertices xx and yy in a graph GG are adjacent, or neighbours, if x,yhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairxy is an edge of GG.


  • affinely independent de

    We call a set (bsetst[i]ui)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstiOMFOREIGNscala.xml.Node$@6040b6ef points in nhttp://mathhub.info/smglom/linear-algebra/nspace.omdoc?nspace?Euclidean-spacen affinely independent, iff the set (bsetst[i]u0-ui)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstihttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6ef are linearly independent.


  • aliquot sequence de

    An aliquot sequence starting with a positive integer kk is a recursive sequence in which each term is the sum of the proper divisors of the previous term.


  • aliquot sum

    A perfect number is a positive integer that is equal to the sum of its aliquot sum.


  • aliquot sum Show Notations

    The aliquot sum s(n)http://mathhub.info/smglom/smglom/aliquotsum.omdoc?aliquotsum?aliquot-sumn is defined as the sum of the proper divisors of nn.


  • all-Harshad number de

    An all-Harshad number or an all-Niven number is a number which is a Harshad number in any number base.

    There are only four all-Harshad numbers: 1, 2, 4, and 6.


  • all-Niven number de

    An all-Harshad number or an all-Niven number is a number which is a Harshad number in any number base.

    There are only four all-Harshad numbers: 1, 2, 4, and 6.


  • alphabetic variant de

    We call a formula AA an alphabetic variant of BB (or http://mathhub.info/smglom/smglom/alpharenaming.omdoc?alpharenaming?alphaeqRel-equal; write (alphaeqA)Bhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/alpharenaming.omdoc?alpharenaming?alphaeqAB), iff BB can be obtained from AA by systematically renaming bound variables.


  • alternating harmonic series de

    The alternating harmonic series is the series

    (infinite-sum1[n](multi-relation-expression((1)(n+1))nequal((((1-(12))+(13))-(14))+(15))-equalln(2)))http://mathhub.info/smglom/calculus/infinitesum.omdoc?infinitesum?infinite-sumnhttp://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnnhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/calculus/naturallogarithm.omdoc?naturallogarithm?natural-logarithm

  • Amicable number de

    Amicable numbers are two different numbers so related that the sum of the divisors (without the numbers themselves) of each is equal to the other number.

    The smallest pair of amicable numbers is 220,284http://mathhub.info/smglom/sets/pair.omdoc?pair?pair.


  • angle Show Notations

    A pair of rays ABhttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?rayAB and AChttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?rayAC is called an angle with vertex AA. We write (AB),(AC)http://mathhub.info/smglom/sets/pair.omdoc?pair?pairhttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?rayABhttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?rayAC as ABChttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?angleABC.


  • anti-reflexive de

    A relation R(A×A)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetRhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAA is called

    • reflexive on AA, iff (a,a)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairaaR for all aAhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaA, and

    • irreflexive (or anti-reflexive) on AA, iff (a,a)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?ninsethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairaaR for all aAhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaA.


  • antisymmetric

    A relation R(A×A)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetRhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAA is called

    • symmetric on AA, iff (b,a)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairbaR for all a,bAhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabA with (a,b)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairabR.

    • asymmetric on AA, iff (b,a)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?ninsethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairbaR for all a,bAhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabA with (a,b)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairabR.

    • antisymmetric on AA, iff (a,b)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairabR and (b,a)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairbaR imply a=bhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalab.


  • approximately equal Show Notations de

    We call two mathematical objects aa and bb approximately equal, (written abhttp://mathhub.info/smglom/mv/approxeq.omdoc?approxeq?approximately-equalab), iff they are discerned only by properties of less relevance in the current situation.


  • arc de

    A graph is a pair V,Ehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE such that VV is a set and E(V×V)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetEhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsVV is a subset of the set of pairs from VV. We call VV the vertices (or nodes, points,junctions) and EE the edges (or lines, branches, arcs) of GG.


  • arithmetic mean Show Notations de

    The arithmetic mean of a set M=({(a1,,an)})http://mathhub.info/smglom/mv/equal.omdoc?equal?equalMhttp://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlian is given by AMhttp://mathhub.info/smglom/numberfields/arithmeticmean.omdoc?arithmeticmean?arithmetic-meanM.


  • ARS

    An abstract reduction system (or abstract rewriting system, or ARS) A,Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureAR consists of a set AA together with a relation R(A2)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetRhttp://mathhub.info/smglom/sets/pair.omdoc?pair?twodimA. The relation RR is written as (arsRconvR)http://mathhub.info/smglom/smglom/abstract-reduction-system.omdoc?abstract-reduction-system?arsRconvR or simply as arsconvhttp://mathhub.info/smglom/smglom/abstract-reduction-system.omdoc?abstract-reduction-system?arsconv.

    The transitive-reflexive closure of arsconvhttp://mathhub.info/smglom/smglom/abstract-reduction-system.omdoc?abstract-reduction-system?arsconv is denoted by arsconvtrhttp://mathhub.info/smglom/smglom/abstract-reduction-system.omdoc?abstract-reduction-system?arsconvtr. To distinguish the original relation arsconvhttp://mathhub.info/smglom/smglom/abstract-reduction-system.omdoc?abstract-reduction-system?arsconv it is sometimes written as arsconvhttp://mathhub.info/smglom/smglom/abstract-reduction-system.omdoc?abstract-reduction-system?arsconv.


  • asymmetric de

    A relation R(A×A)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetRhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAA is called

    • symmetric on AA, iff (b,a)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairbaR for all a,bAhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabA with (a,b)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairabR.

    • asymmetric on AA, iff (b,a)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?ninsethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairbaR for all a,bAhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabA with (a,b)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairabR.

    • antisymmetric on AA, iff (a,b)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairabR and (b,a)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairbaR imply a=bhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalab.


  • asymptotic density Show Notations de

    A subset AA of positive integers has an asymptotic density (or natural density) d(A)http://mathhub.info/smglom/smglom/asymptoticdensity.omdoc?asymptoticdensity?asymptotic-densityA, where (multi-relation-expression0lessthand(A)lessthan1)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthanhttp://mathhub.info/smglom/smglom/asymptoticdensity.omdoc?asymptoticdensity?asymptotic-densityAhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthan, if the limit exists

    d(A)http://mathhub.info/smglom/smglom/asymptoticdensity.omdoc?asymptoticdensity?asymptotic-densityA

    anhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationan is the number of elements of AA less than or equal to nn.


  • Bailey-Borwein-Plouffe formula de

    The Bailey-Borwein-Plouffe formula (BBP formula) is a series for the computation of πhttp://mathhub.info/smglom/numberfields/pinumber.omdoc?pinumber?pinumber:

    πhttp://mathhub.info/smglom/numberfields/pinumber.omdoc?pinumber?pinumber

  • balanced prime de

    A balanced prime is a prime number that is equal to the arithmetic mean of the nearest primes above and below.

    (pn)=(((p(n-1))+(p(n+1)))2)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?prime-numbernhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?prime-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?prime-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionn

    where pnhttp://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?prime-numbern is the nnth prime number.


  • barrier de

    A number nn is called a barrier of a number-theoretic function fmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationfm if, for all m>nhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanmn, (m+(fm))<nhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthanhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationfmn.


  • base set de

    Let MM be a set, then we call a function d:(M×M)http://mathhub.info/smglom/sets/functions.omdoc?functions?fundhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsMMhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number a distance function (or metric ) on MM, iff for all x,y,zMhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetxyzM the following three identities hold:

    1. 1.

      (d)=0http://mathhub.info/smglom/mv/equal.omdoc?equal?equald iff x=yhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalxy (identity of indiscernibles),

    2. 2.

      (d)=(d)http://mathhub.info/smglom/mv/equal.omdoc?equal?equaldd (symmetry), and

    3. 3.

      (d)((d)+(d))http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethandhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additiondd (triangle inequality).

    We call M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd a metric space with base set MM and metric dd.


  • base-b repunits

    A repunit is a natural number that contains only the digit 1.

    The base-b repunits are defined as (Rn(b))=(((bn)-1)(b-1))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/repunit.omdoc?repunit?repunitbnbhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationbnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionb for b2http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methanb and n1http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methann.

    A repunit prime is a repunit that is also a prime number.


  • Baxter-Hickerson function

    The Baxter-Hickerson function is defined for non-negative integers nn by

    (fn)=((13)(((210(5n))-(10(4n)))+(210(3n))+(10(2n))+(10n)+1))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationfnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn

    It produces numbers whose cubes are zerofree.


  • BBP formula de

    The Bailey-Borwein-Plouffe formula (BBP formula) is a series for the computation of πhttp://mathhub.info/smglom/numberfields/pinumber.omdoc?pinumber?pinumber:

    πhttp://mathhub.info/smglom/numberfields/pinumber.omdoc?pinumber?pinumber

  • between Show Notations

    We call a set of points a ternary relation betweenhttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?between an order geometry , iff

    1. 1.

      If A*B*Chttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?betweenABC, then

      1. (a)

        AChttp://mathhub.info/smglom/mv/equal.omdoc?equal?notequalAC.

      2. (b)

        {A,B,C}http://mathhub.info/smglom/sets/set.omdoc?set?setABC is a collinear set.

      3. (c)

        B*B*Ahttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?betweenBBA.

    2. 2.

      If BB and DD are distinct points, then there is at least one point AA, such that A*B*Dhttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?betweenABD.

    3. 3.

      If AA, BB and CC are distinct collinear points, then A*B*Chttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?betweenABC, or A*C*Bhttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?betweenACB, or B*A*Chttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?betweenBAC.

    4. 4.

      A*B*Chttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?betweenABC and A*C*Bhttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?betweenACB cannot hold at the same time.

    If A*B*Chttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?betweenABC, we say that BB is between AA and CC.


  • bijective de

    A function f:SThttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfST is called bijective, iff ff is injective and surjective.


  • binary logarithm de

    The binary logarithm is the logarithm to the base 2.


  • binomial coefficient Show Notations de

    The binomial coefficient 𝒞knhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficientnk (written as 𝒞knhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficientnk in French) is defined to be the number of kk-element subsets of an nn-element set.


  • biprime de

  • branch de

    A graph is a pair V,Ehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE such that VV is a set and E(V×V)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetEhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsVV is a subset of the set of pairs from VV. We call VV the vertices (or nodes, points,junctions) and EE the edges (or lines, branches, arcs) of GG.


  • Briggsian logarithm Show Notations de

    The common logarithm is the logarithm to the base 10. It is also known as the decadic logarithm and also as the decimal logarithm, named after its base, or Briggsian logarithm after Henry Briggs who introduced it.


  • cabtaxi number Show Notations de

    The nnth cabtaxi number Cabtaxi(n)http://mathhub.info/smglom/numbers/cabtaxinumber.omdoc?cabtaxinumber?cabtaxi-numbern is defined as the smallest positive integer that can be expressed as a sum or a difference of two cubes (included 0) in nn distinct ways.

    For example:

    (multi-relation-expressionCabtaxi(2)equal91equal(33)+(43)equal(63)+(53))http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numbers/cabtaxinumber.omdoc?cabtaxinumber?cabtaxi-numberhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiation

  • Cahen’s constant Show Notations de

    Let snhttp://mathhub.info/smglom/numbers/sylvestersequence.omdoc?sylvestersequence?sylvestersequencen be the Sylvester sequence.

    Cahen’s constant is defined as an infinite series of unit fractions with alternating signs

    𝐶:=(infinite-sum0[i]((1)i)((si)-1))http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/smglom/cahenconstant.omdoc?cahenconstant?cahenconstanthttp://mathhub.info/smglom/calculus/infinitesum.omdoc?infinitesum?infinite-sumihttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionihttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numbers/sylvestersequence.omdoc?sylvestersequence?sylvestersequencei

  • cake number de

    The cake number Cnhttp://mathhub.info/smglom/numbers/cakenumber.omdoc?cakenumber?cakenumbern is the maximum number of regions into which a 3-dimensional cube can be partitioned by exactly nn planes.

    (multi-relation-expressionCnequal(𝒞3n)+(𝒞2n)+(𝒞1n)+(𝒞0n)equal(16)((n3)+(5n)+6))http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numbers/cakenumber.omdoc?cakenumber?cakenumbernhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficientnhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficientnhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficientnhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficientnhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationn

  • cardinality Show Notations de

    We say that a set AA is finite and has cardinality (or size) (#(A))http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityAhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, iff there is a bijective function f:A({n|(n<(#(A)))})http://mathhub.info/smglom/sets/functions.omdoc?functions?funfAhttp://mathhub.info/smglom/sets/set.omdoc?set?rsetsthttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-numbernhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthannhttp://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityA.

    The cardinality of a set AA is also written as #(A)http://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityA, #(A)http://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityA, #(A)http://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityA, or #(A)http://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityA.


  • Carol number de

    A Carol number is an integer of the form

    ((4n)-(2(n+1))-1)=((((2n)-1)2)-2)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn

    for a positive integer nn.


  • Cassini identity de

  • Catalan identity de

  • Catalan number de

    The Catalan numbers form a sequence of natural numbers defind by

    (multi-relation-expressionCnequal(𝒞n(2n))-(𝒞(n+1)(2n))equal(1(n+1))(𝒞n(2n))equal((2n)!)((n+1)!))http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numbers/catalannumber.omdoc?catalannumber?catalannumbernhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficienthttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnnhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficienthttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficienthttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnnhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/factorial.omdoc?factorial?factorialhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnhttp://mathhub.info/smglom/numberfields/factorial.omdoc?factorial?factorialhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionn

    for n0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methann.


  • Catalan-Mersenne number de

  • Cauchy sequence de

    Let M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd be a metric space, then we call a sequence (sequenceon[name.cvar.0]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonname.cvar.0n a Cauchy sequence, iff for each ϵ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanϵ there exists a n0http://mathhub.info/smglom/sets/set.omdoc?set?insetOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, such that (d)<ϵhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthandϵ for all n,mn0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?mmethannmOMFOREIGNscala.xml.Node$@6040b6ef.


  • ceiling Show Notations de

    The floor

    r
    http://mathhub.info/smglom/numberfields/ceilingfloor.omdoc?ceilingfloor?floorr of a real number rr is the largest integer that is smaller or equal to rr. The floor function is also called the greatest integer function.

    The ceiling ]]r[[http://mathhub.info/smglom/numberfields/ceilingfloor.omdoc?ceilingfloor?ceilingr of a real number rr is the smallest integer that is greater or equal to rr. The ceiling function is also called the least integer function.


  • central Delannoy number Show Notations de

    The central Delannoy numbers (D(n))=(D(n,n))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/centraldelannoynumber.omdoc?centraldelannoynumber?central-Delannoy-numbernhttp://mathhub.info/smglom/numbers/delannoynumber.omdoc?delannoynumber?Delannoy-numbernn are the Delannoy numbers for a square nnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnn grid.

    The first central Delannoy numbers are:

    1,3,13,63,321,1683,8989,48639,265729,...http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?infseq

  • central polygonal number de

    The central polygonal numbers or lazy caterer’s sequence describes the maximum number pp of pieces of a circle (or a plane) that can be made with a given number nn of straight cuts.

    (multi-relation-expressionpequal(𝒞2n)+(𝒞1n)+(𝒞0n)equal((n2)+n+2)2)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionphttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficientnhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficientnhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficientnhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationnn

  • Chebyshev function de

    The summatory von Mangoldt function, also known as the Chebyshev function, is defined as

    ψψ

  • child

    A tree is a directed acyclic graph G=(V,E)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalGhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE such that

    • there is exactly one initialnode vrVhttp://mathhub.info/smglom/sets/set.omdoc?set?insetOMFOREIGNscala.xml.Node$@6040b6efV (called the root), and

    • all nodes but the root have indegree 1.

    We call vv the parent of ww, iff (v,w)Ehttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairvwE (ww is a child of vv). We call a node vv a leaf of GG, iff it is terminal, i.e. if it does not have children.


  • chord de

    If in a graph GG which contains a cycle CC, an edge of GG joins two vertices of CC but is not itself an edge of the cycle, then this edge is a chord of the cycle CC. Thus a cycle is chordless iff it is an induced cycle in GG.


  • chordless de

    If in a graph GG which contains a cycle CC, an edge of GG joins two vertices of CC but is not itself an edge of the cycle, then this edge is a chord of the cycle CC. Thus a cycle is chordless iff it is an induced cycle in GG.


  • circular prime de

    A prime number is a circular prime if all numbers generated by cyclically permuting of its digits are prime.

    For example, 197 is a circular prime, because 971 and 719 also primes.


  • circumference de

    For a graph GG, the minimum length of a cycle contained in GG is the girth g(G)http://mathhub.info/smglom/graphs/graphcycleparameters.omdoc?graphcycleparameters?girthG of GG. The maximum length of a cycle contained in GG is the circumference of GG. For a graph which does not contain a cycle the girth is set to (g(G))=http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/graphs/graphcycleparameters.omdoc?graphcycleparameters?girthGhttp://mathhub.info/smglom/numberfields/infinity.omdoc?infinity?infinity, its circumference is set to zero.


  • clopen set

    A topological space X,Ohttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureXOMFOREIGNscala.xml.Node$@6040b6ef is a set XX together with a collection O(𝒫(X))http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/sets/powerset.omdoc?powerset?powersetX, such that

    1. 1.

      ,XOhttp://mathhub.info/smglom/sets/set.omdoc?set?minsethttp://mathhub.info/smglom/sets/emptyset.omdoc?emptyset?esetXOMFOREIGNscala.xml.Node$@6040b6ef.

    2. 2.

      (SSs)Ohttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/union.omdoc?union?munionCollectionOMFOREIGNscala.xml.Node$@6040b6efSsOMFOREIGNscala.xml.Node$@6040b6ef if SShttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetSOMFOREIGNscala.xml.Node$@6040b6ef.

    3. 3.

      (SSs)Ohttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/intersection.omdoc?intersection?mintersectCollectionOMFOREIGNscala.xml.Node$@6040b6efSsOMFOREIGNscala.xml.Node$@6040b6ef if SShttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetSOMFOREIGNscala.xml.Node$@6040b6ef is finite.

    OOMFOREIGNscala.xml.Node$@6040b6ef is called an open set topology (or just topology) on XX. Members of a topology OOMFOREIGNscala.xml.Node$@6040b6ef are called open sets and their complements closed sets. A subset of XX may be neither closed nor open, either closed or open, or both. A set that is both closed and open is called a clopen set.


  • closed ball Show Notations de

    Let M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd be a metric space, then we call the set (fundefeq[rx]𝐵(r,x))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqrxhttp://mathhub.info/smglom/calculus/ball.omdoc?ball?open-ballrx the open ball and (fundefeq[rx]B¯(r,x))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqrxhttp://mathhub.info/smglom/calculus/ball.omdoc?ball?closed-ballrx the closed ball around xx with radius rr. We also write 𝐵(r,x)http://mathhub.info/smglom/calculus/ball.omdoc?ball?open-ballrx and B¯(r,x)http://mathhub.info/smglom/calculus/ball.omdoc?ball?closed-ballrx.


  • closed set de

    A topological space X,Ohttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureXOMFOREIGNscala.xml.Node$@6040b6ef is a set XX together with a collection O(𝒫(X))http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/sets/powerset.omdoc?powerset?powersetX, such that

    1. 1.

      ,XOhttp://mathhub.info/smglom/sets/set.omdoc?set?minsethttp://mathhub.info/smglom/sets/emptyset.omdoc?emptyset?esetXOMFOREIGNscala.xml.Node$@6040b6ef.

    2. 2.

      (SSs)Ohttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/union.omdoc?union?munionCollectionOMFOREIGNscala.xml.Node$@6040b6efSsOMFOREIGNscala.xml.Node$@6040b6ef if SShttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetSOMFOREIGNscala.xml.Node$@6040b6ef.

    3. 3.

      (SSs)Ohttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/intersection.omdoc?intersection?mintersectCollectionOMFOREIGNscala.xml.Node$@6040b6efSsOMFOREIGNscala.xml.Node$@6040b6ef if SShttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetSOMFOREIGNscala.xml.Node$@6040b6ef is finite.

    OOMFOREIGNscala.xml.Node$@6040b6ef is called an open set topology (or just topology) on XX. Members of a topology OOMFOREIGNscala.xml.Node$@6040b6ef are called open sets and their complements closed sets. A subset of XX may be neither closed nor open, either closed or open, or both. A set that is both closed and open is called a clopen set.


  • co-prime Show Notations de

    Two integers are said to be coprime (also spelled co-prime), relatively prime or mutually prime if their greatest common divisor is 1.


  • codomain Show Notations de

    A relation f(X×Y)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetfhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsXY, is called a partial function with domain XX (write 𝐝𝐨𝐦(f)http://mathhub.info/smglom/sets/functions.omdoc?functions?domainf) and codomain YY (write 𝐜𝐨𝐝𝐨𝐦(f)http://mathhub.info/smglom/sets/functions.omdoc?functions?codomainf), iff for all xXhttp://mathhub.info/smglom/sets/set.omdoc?set?insetxX there is at most one yYhttp://mathhub.info/smglom/sets/set.omdoc?set?insetyY with (x,y)fhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairxyf.

    We write f:XY;xyhttp://mathhub.info/smglom/sets/functions.omdoc?functions?partfunsuchthatfXYxy and (f)=yhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalfy instead of (x,y)fhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairxyf.


  • collinear

    A set SS of points is called collinear, if there is a line ll with (p,l)Ihttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairplI for all sShttp://mathhub.info/smglom/sets/set.omdoc?set?insetsS.


  • common logarithm Show Notations de

    The common logarithm is the logarithm to the base 10. It is also known as the decadic logarithm and also as the decimal logarithm, named after its base, or Briggsian logarithm after Henry Briggs who introduced it.


  • commutative de

    A ring Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR is called commutative if Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR is a commutative magma.


  • commutator Show Notations de

    Let Ghttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureG be a group and a,bGhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabG. Then we define the commutator of aa and bb as (fundefeq[ab][a,b])http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqabhttp://mathhub.info/smglom/algebra/commutator.omdoc?commutator?commutatorab. It is equal to the group’s unit if and only if gg and hh commute.


  • compact

    Let X,Ohttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureXOMFOREIGNscala.xml.Node$@6040b6ef be a topological space and KXhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetKX, then we call KK compact, iff every cover of KK has a finite subcover.


  • complete de

    A metric space M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd is called complete (or a complete space), iff every Cauchy sequence in M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd is convergent.


  • complete de

    If in a graph GG all vertices are pairwise adjacent, then we call GG complete. A complete graph on nn vertices is denoted by Knhttp://mathhub.info/smglom/graphs/graphcomplete.omdoc?graphcomplete?completegraphn.


  • complete graph Show Notations de

    If in a graph GG all vertices are pairwise adjacent, then we call GG complete. A complete graph on nn vertices is denoted by Knhttp://mathhub.info/smglom/graphs/graphcomplete.omdoc?graphcomplete?completegraphn.


  • complete space de

    A metric space M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd is called complete (or a complete space), iff every Cauchy sequence in M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd is convergent.


  • complex number Show Notations de

    The set http://mathhub.info/smglom/numberfields/complexnumbers.omdoc?complexnumbers?complex-number of complex numbers contains numbers of the form a+(b𝑖)http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionahttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationbhttp://mathhub.info/smglom/numberfields/complexnumbers.omdoc?complexnumbers?imaginary-unit, where a,bhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number. We call 𝑖http://mathhub.info/smglom/numberfields/complexnumbers.omdoc?complexnumbers?imaginary-unit the imaginary unit.


  • component de

    A structure combines multiple mathematical objects (the components) into a new object. Structures are usually given as finite enumerations, where the components have names by which they can be referenced.


  • composite number de

    A composite number is a positive integer that has at least one positive divisor other than 1 or itself.


  • composition Show Notations de

    The composition of two relations R(A×B)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetRhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAB and S(B×C)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetShttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsBC is defined as (fundefeq[SR]SR)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqSRhttp://mathhub.info/smglom/sets/relation-composition.omdoc?relation-composition?compositionSR


  • connected graph de

    A non-empty graph GG is said to be a connected graph if any two of its vertices are linked by a path in GG. If the subgraph GUhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationGU, induced by a subset UU of the vertex set VGhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationVG, is connected, we call UU itself connected (in GG). For the negation we usually prefer ’disconnected’ over ’not connected’.


  • continuous de

    Let X,Ohttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureXOMFOREIGNscala.xml.Node$@6040b6ef and X,Ohttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6ef be topological spaces, then we call a function f:XYhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfXY continuous, iff ((𝐈𝐦(f))X)Ohttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/sets/image.omdoc?image?imagefXOMFOREIGNscala.xml.Node$@6040b6ef for all XOhttp://mathhub.info/smglom/sets/set.omdoc?set?insetXOMFOREIGNscala.xml.Node$@6040b6ef.


  • convergent de

    We say that a sequence is convergent, iff it converges against a limit gg.


  • converges de

    Let M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd be a metric space and (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan a sequence with (ai)Mhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselaiM for all ihttp://mathhub.info/smglom/sets/set.omdoc?set?insetihttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, then we say that (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan converges to gMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetgM (we call gg the limit of (na)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequencean and write limnanhttp://mathhub.info/smglom/calculus/sequenceConvergence.omdoc?sequenceConvergence?limitnOMFOREIGNscala.xml.Node$@6040b6ef ), iff for each ϵ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanϵ there exists a n0http://mathhub.info/smglom/sets/set.omdoc?set?insetOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, such that (d)<ϵhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthandϵ for all nn0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methannOMFOREIGNscala.xml.Node$@6040b6ef.


  • converges de

    Let X,Ohttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureXOMFOREIGNscala.xml.Node$@6040b6ef be a Hausdorff space, then we say that a sequence (sequenceon[name.cvar.0]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonname.cvar.0n converges to xXhttp://mathhub.info/smglom/sets/set.omdoc?set?insetxX (we call xx the Hausdorff limit and write xx), iff every neighborhood of xx only contains finitely many xiOMFOREIGNscala.xml.Node$@6040b6ef.


  • converse relation Show Notations de

    (fundefeq[R]R-1)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqRhttp://mathhub.info/smglom/sets/converse-relation.omdoc?converse-relation?converse-relationR is the converse relation of R(A×B)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetRhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAB.


  • convex de

    A set S(n)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetShttp://mathhub.info/smglom/linear-algebra/nspace.omdoc?nspace?Euclidean-spacen is called convex, iff (bsetst[t](ta)+((1-t)b))Shttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsethttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstthttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationtahttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractiontbS for all a,bShttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabS.


  • convex hull de

  • convex polytope de

  • coprime Show Notations de

    Two integers are said to be coprime (also spelled co-prime), relatively prime or mutually prime if their greatest common divisor is 1.


  • cosine integral Show Notations de

    The cosine integral Cin(x)http://mathhub.info/smglom/smglom/cosineintegralint.omdoc?cosineintegralint?cosine-integralx is defined by:

    Cin(x)http://mathhub.info/smglom/smglom/cosineintegralint.omdoc?cosineintegralint?cosine-integralx

  • countable de

    We say that a set AA is countable, iff there is a surjective function f:Ahttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfAhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number.


  • countably infinite de

    We say that a set AA is countably infinite, iff there is a bijective function f:Ahttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfAhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number.


  • Cousin prime de

    Cousin primes are prime numbers that differ from each other by four.


  • cover de

    A cover of a set XX is a collection CC of sets, such that X(aAUa)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetXhttp://mathhub.info/smglom/sets/union.omdoc?union?munionCollectionAaOMFOREIGNscala.xml.Node$@6040b6ef. A subset CChttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efC is called a subcover of XX, iff it still covers XX.


  • cross product Show Notations de

    The cross product of two vectors in 3http://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?nCartSpacehttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number is their vector product.


  • cubefree taxicab number de

  • cubic graph de

    A 3-regular graph is called cubic graph.


  • Cullen number Show Notations de

    A Cullen number is a number of the form (Cn)=((n(2n))+1)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/cullennumber.omdoc?cullennumber?Cullen-numbernhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn


  • Cullen numbers of the second kind Show Notations de

  • cycle de

    Given a directed graph G=(V,E)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalGhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE,

    • a path pp is called cyclic (or a cycle) iff (start(p))=(end(p))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pstartphttp://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pendp.

    • a cycle (v0,,vn)http://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlivn is called simple, iff vivjhttp://mathhub.info/smglom/mv/equal.omdoc?equal?notequalOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6ef for i,j1nhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?mbetweeneeijn with ijhttp://mathhub.info/smglom/mv/equal.omdoc?equal?notequalij.

    • GG is called acyclic (or a DAG (directed acyclic graph)) iff there is no cycle in GG.


  • cyclic de

    Given a directed graph G=(V,E)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalGhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE,

    • a path pp is called cyclic (or a cycle) iff (start(p))=(end(p))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pstartphttp://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pendp.

    • a cycle (v0,,vn)http://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlivn is called simple, iff vivjhttp://mathhub.info/smglom/mv/equal.omdoc?equal?notequalOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6ef for i,j1nhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?mbetweeneeijn with ijhttp://mathhub.info/smglom/mv/equal.omdoc?equal?notequalij.

    • GG is called acyclic (or a DAG (directed acyclic graph)) iff there is no cycle in GG.


  • cyclic number de

    A cyclic number is a natural number in which cyclic permutations of the digits are successive multiples of the number.

    (1428571)=142857http://mathhub.info/smglom/mv/equal.omdoc?equal?equal
    (1428572)=285714http://mathhub.info/smglom/mv/equal.omdoc?equal?equal
    (1428573)=428571http://mathhub.info/smglom/mv/equal.omdoc?equal?equal
    (1428574)=571428http://mathhub.info/smglom/mv/equal.omdoc?equal?equal
    (1428575)=714285http://mathhub.info/smglom/mv/equal.omdoc?equal?equal
    (1428576)=857142http://mathhub.info/smglom/mv/equal.omdoc?equal?equal

  • DAG

    Given a directed graph G=(V,E)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalGhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE,

    • a path pp is called cyclic (or a cycle) iff (start(p))=(end(p))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pstartphttp://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pendp.

    • a cycle (v0,,vn)http://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlivn is called simple, iff vivjhttp://mathhub.info/smglom/mv/equal.omdoc?equal?notequalOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6ef for i,j1nhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?mbetweeneeijn with ijhttp://mathhub.info/smglom/mv/equal.omdoc?equal?notequalij.

    • GG is called acyclic (or a DAG (directed acyclic graph)) iff there is no cycle in GG.


  • decadic logarithm Show Notations de

    The common logarithm is the logarithm to the base 10. It is also known as the decadic logarithm and also as the decimal logarithm, named after its base, or Briggsian logarithm after Henry Briggs who introduced it.


  • decimal logarithm Show Notations de

    The common logarithm is the logarithm to the base 10. It is also known as the decadic logarithm and also as the decimal logarithm, named after its base, or Briggsian logarithm after Henry Briggs who introduced it.


  • defined piecewise Show Notations de

    A function mm is defined piecewise, we write

    (mx)=(defined-piecewise(
    a1wennA1
    )
    (
    wenn
    )
    (
    anwennAn
    )
    (
    osonst
    )
    )
    http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationmxhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?pieceOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?pieceOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwiseo

    where AiOMFOREIGNscala.xml.Node$@6040b6ef are conditions involving xx, if (mx)=aihttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationmxOMFOREIGNscala.xml.Node$@6040b6ef for all xx with AiOMFOREIGNscala.xml.Node$@6040b6ef and oo otherwise.


  • definiendum de

    If aa does not occur in AA, we call a pair a:=Ahttp://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationaA a definitional equation with definiendum aa and definiens AA.


  • definiens de

    If aa does not occur in AA, we call a pair a:=Ahttp://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationaA a definitional equation with definiendum aa and definiens AA.


  • definite integral Show Notations

    Given a function f:http://mathhub.info/smglom/sets/functions.omdoc?functions?funfhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-numberhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number and an interval ([a,b])http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?proper-subsethttp://mathhub.info/smglom/calculus/interval.omdoc?interval?ccintervalabhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number, then the definite integral (definite-integralab[fx]x)http://mathhub.info/smglom/calculus/definiteintegral.omdoc?definiteintegral?definite-integralabfxx is defined to be the signed area of the region in the plane bounded by the graph of ff, the xx-axis, and the vertical lines x=ahttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalxa and x=bhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalxb, such that area above the xx-axis adds to the total, and that below the xx-axis subtracts from the total.


  • definitional equation Show Notations de

    If aa does not occur in AA, we call a pair a:=Ahttp://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationaA a definitional equation with definiendum aa and definiens AA.


  • Delannoy number Show Notations de

    A Delannoy number D(m,n)http://mathhub.info/smglom/numbers/delannoynumber.omdoc?delannoynumber?Delannoy-numbermn describes the number of paths from the southwest corner 0,0http://mathhub.info/smglom/sets/pair.omdoc?pair?pair of a rectangular grid to the northeast corner m,nhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairmn, using only single steps north, northeast, or east.

    It follows that (multi-relation-expressionD(0,n)equalD(m,0)equal1)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numbers/delannoynumber.omdoc?delannoynumber?Delannoy-numbernhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/delannoynumber.omdoc?delannoynumber?Delannoy-numbermhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal and (D(m,n))=((D((m-1),n))+(D((m-1),(n-1)))+(D(m,(n-1))))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/delannoynumber.omdoc?delannoynumber?Delannoy-numbermnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numbers/delannoynumber.omdoc?delannoynumber?Delannoy-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionmnhttp://mathhub.info/smglom/numbers/delannoynumber.omdoc?delannoynumber?Delannoy-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numbers/delannoynumber.omdoc?delannoynumber?Delannoy-numbermhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn for all (mnotequalmn0)http://mathhub.info/smglom/mv/equal.omdoc?equal?mnotequalmn


  • derivative Show Notations de

    If f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN is differentiable on MM, then the derivative function (Dxf):MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfxMN of ff (with respect to xx) is defined as

    (fundefeq[fxa]Dxf(a))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfxahttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa

    The dependency of Dxfhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfx on xx is left implicit in this notation (Lagrange’s notation). In Leibniz’s notation we write the derivative of ff with respect to xx as Dxfhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfx and the derivative of ff at aa as Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa or Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa. In Euler’s notation, this is written as Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa and in Newton’snotation as Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa.

    We call f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN differentiable at a limit point pMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetpM, iff

    d:=(limxp((dB)(dA)))http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationdhttp://mathhub.info/smglom/calculus/functionlimit.omdoc?functionlimit?limitpxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6ef

    exists; in this case we say that ff has derivative dd at pp. We say that ff is differentiable on MM, iff it is differentiable at every mMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetmM.


  • derivative function de

    If f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN is differentiable on MM, then the derivative function (Dxf):MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfxMN of ff (with respect to xx) is defined as

    (fundefeq[fxa]Dxf(a))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfxahttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa

    The dependency of Dxfhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfx on xx is left implicit in this notation (Lagrange’s notation). In Leibniz’s notation we write the derivative of ff with respect to xx as Dxfhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfx and the derivative of ff at aa as Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa or Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa. In Euler’s notation, this is written as Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa and in Newton’snotation as Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa.


  • differentiable de

    If f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN is differentiable on MM, then the derivative function (Dxf):MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfxMN of ff (with respect to xx) is defined as

    (fundefeq[fxa]Dxf(a))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfxahttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa

    The dependency of Dxfhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfx on xx is left implicit in this notation (Lagrange’s notation). In Leibniz’s notation we write the derivative of ff with respect to xx as Dxfhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfx and the derivative of ff at aa as Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa or Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa. In Euler’s notation, this is written as Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa and in Newton’snotation as Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa.

    We call f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN differentiable at a limit point pMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetpM, iff

    d:=(limxp((dB)(dA)))http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationdhttp://mathhub.info/smglom/calculus/functionlimit.omdoc?functionlimit?limitpxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6ef

    exists; in this case we say that ff has derivative dd at pp. We say that ff is differentiable on MM, iff it is differentiable at every mMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetmM.


  • differentiable on de

    If f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN is differentiable on MM, then the derivative function (Dxf):MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfxMN of ff (with respect to xx) is defined as

    (fundefeq[fxa]Dxf(a))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfxahttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa

    The dependency of Dxfhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfx on xx is left implicit in this notation (Lagrange’s notation). In Leibniz’s notation we write the derivative of ff with respect to xx as Dxfhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfx and the derivative of ff at aa as Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa or Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa. In Euler’s notation, this is written as Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa and in Newton’snotation as Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa.

    We call f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN differentiable at a limit point pMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetpM, iff

    d:=(limxp((dB)(dA)))http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationdhttp://mathhub.info/smglom/calculus/functionlimit.omdoc?functionlimit?limitpxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6ef

    exists; in this case we say that ff has derivative dd at pp. We say that ff is differentiable on MM, iff it is differentiable at every mMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetmM.


  • digit product de

    The digit product of a given natural number is the product of all its digits.


  • digit sum de

    The digit sum of a given natural number is the sum of all its digits.


  • digital root

    The number of times the digits must be summed to reach the digital sum is called a number’s additive persistence.


  • digraph de

    A directed graph (also called digraph or oriented graph) is a pair V,Ehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE such that VV is a set and E(V×V)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetEhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsVV. We call VV the vertices (or nodes) and EE the edges of GG.


  • Diophantine equation de

    A Diophantine equation is a polynomial equation that allows the variables to take integer values only.


  • directed graph de

    A directed graph (also called digraph or oriented graph) is a pair V,Ehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE such that VV is a set and E(V×V)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetEhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsVV. We call VV the vertices (or nodes) and EE the edges of GG.


  • directed set

    If lessthanhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthan is a preorder on a set DD, then Dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureD is called a directed set, iff all a,bDhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabD have an upper bound cDhttp://mathhub.info/smglom/sets/set.omdoc?set?insetcD .


  • disconnected

    A non-empty graph GG is said to be a connected graph if any two of its vertices are linked by a path in GG. If the subgraph GUhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationGU, induced by a subset UU of the vertex set VGhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationVG, is connected, we call UU itself connected (in GG). For the negation we usually prefer ’disconnected’ over ’not connected’.


  • disjoint de ro tr

    Two sets AA and BB are called disjoint, iff (AB)=http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/sets/intersection.omdoc?intersection?intersectionABhttp://mathhub.info/smglom/sets/emptyset.omdoc?emptyset?eset.

    A family of sets is called pairwise disjoint or mutually disjoint, if any two of them are disjoint.


  • distance function de

    Let MM be a set, then we call a function d:(M×M)http://mathhub.info/smglom/sets/functions.omdoc?functions?fundhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsMMhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number a distance function (or metric ) on MM, iff for all x,y,zMhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetxyzM the following three identities hold:

    1. 1.

      (d)=0http://mathhub.info/smglom/mv/equal.omdoc?equal?equald iff x=yhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalxy (identity of indiscernibles),

    2. 2.

      (d)=(d)http://mathhub.info/smglom/mv/equal.omdoc?equal?equaldd (symmetry), and

    3. 3.

      (d)((d)+(d))http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethandhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additiondd (triangle inequality).

    We call M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd a metric space with base set MM and metric dd.


  • divisor Show Notations de

    Let Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR be a ring and m0http://mathhub.info/smglom/mv/equal.omdoc?equal?notequalm, then we say that mm is a left divisor of nn (or mm left divides nn) and that nn is a left multiple of mm, if there is a kRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetkR such that (multiplication)=nhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationn.

    Analogously we say that mm is a right divisor of nn (or mm right divides nn) and that nn is a right multiple of mm, if there is a kRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetkR such that (multiplication)=nhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationn.

    If RR is commxutative, then left and right divisors coincide and we simply speak of divisor and multiple and write m|nhttp://mathhub.info/smglom/algebra/ring-divisor.omdoc?ring-divisor?divisormn.

    1, 1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction, nn, and nhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn are known as the trivial divisors of nn. A divisor of nn that is not a trivial divisor is known as a proper or non-trivial divisor.


  • domain Show Notations de

    A relation f(X×Y)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetfhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsXY, is called a partial function with domain XX (write 𝐝𝐨𝐦(f)http://mathhub.info/smglom/sets/functions.omdoc?functions?domainf) and codomain YY (write 𝐜𝐨𝐝𝐨𝐦(f)http://mathhub.info/smglom/sets/functions.omdoc?functions?codomainf), iff for all xXhttp://mathhub.info/smglom/sets/set.omdoc?set?insetxX there is at most one yYhttp://mathhub.info/smglom/sets/set.omdoc?set?insetyY with (x,y)fhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairxyf.

    We write f:XY;xyhttp://mathhub.info/smglom/sets/functions.omdoc?functions?partfunsuchthatfXYxy and (f)=yhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalfy instead of (x,y)fhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairxyf.


  • double Mersenne number Show Notations de

  • double Mersenne prime de

    A double Mersenne prime is a double Mersenne number that is prime.


  • edge de

    A directed graph (also called digraph or oriented graph) is a pair V,Ehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE such that VV is a set and E(V×V)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetEhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsVV. We call VV the vertices (or nodes) and EE the edges of GG.


  • edge de

    A graph is a pair V,Ehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE such that VV is a set and E(V×V)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetEhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsVV is a subset of the set of pairs from VV. We call VV the vertices (or nodes, points,junctions) and EE the edges (or lines, branches, arcs) of GG.


  • Egyptian fraction de

  • eighth smarandache constant Show Notations de

    The eighth smarandache constant http://mathhub.info/smglom/smglom/smarandacheconstant8.omdoc?smarandacheconstant8?eighth-smarandache-constant is defind for a natural number kk by

    http://mathhub.info/smglom/smglom/smarandacheconstant8.omdoc?smarandacheconstant8?eighth-smarandache-constant

    where S(n)http://mathhub.info/smglom/smglom/smarandachefunction.omdoc?smarandachefunction?smarandache-functionn is the smarandache function.

    The sum converges.


  • eleventh smarandache constants

    The eleventh smarandache constants eleventh-smarandache-constanthttp://mathhub.info/smglom/smglom/smarandacheconstant11.omdoc?smarandacheconstant11?eleventh-smarandache-constant are defind by

    s11(α)http://mathhub.info/smglom/smglom/smarandacheconstant11.omdoc?smarandacheconstant11?eleventh-smarandache-constantα

    where S(n)http://mathhub.info/smglom/smglom/smarandachefunction.omdoc?smarandachefunction?smarandache-functionn is the smarandache function.

    The sum converges for all real numbers α>1http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanα.


  • emirp de

    An emirp (prime spelled backwards) is a prime number that results in a different prime when its digits are reversed.


  • empty product Show Notations de

    An empty product, or nullary product, is the result of multiplying no factors. It is equal to the multiplicative identity 1.


  • empty set Show Notations tr de ro

    The empty set http://mathhub.info/smglom/sets/emptyset.omdoc?emptyset?eset (also written as http://mathhub.info/smglom/sets/emptyset.omdoc?emptyset?eset) is the set without elements.


  • empty sum Show Notations de

    An empty sum, or nullary sum, is a summation involving no terms at all. The value of any empty sum of numbers is conventionally taken to be zero.

    For example, if nmhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethannm then

    i=mnaihttp://mathhub.info/smglom/numberfields/sum.omdoc?sum?summniOMFOREIGNscala.xml.Node$@6040b6ef

  • end de

    Given a directed graph G=(V,E)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalGhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE we call a vector (p=((v0,,vn)))(V(n+1))http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalphttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlivnhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?nCartSpaceVhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionn a path in GG iff (vi-1,vi)Ehttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6efE for all 1inhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?betweeneein, n>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethann.

    • v0OMFOREIGNscala.xml.Node$@6040b6ef is called the start of pp (write start(p)http://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pstartp )

    • vnOMFOREIGNscala.xml.Node$@6040b6ef is called the end of pp (write end(p)http://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pendp )

    • nn is called the length of pp (write len(p)http://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?plenp )


  • end point

    For distinct points AA and BB, we call the set

    (fundefeq[AB]AB)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqABhttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?rayAB

    the ray from AA through BB. We call AA the end point of ABhttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?rayAB.


  • equal Show Notations de

    We call two mathematical objects aa and bb equal, (written a=bhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalab), iff there are no properties that discern them.


  • equivalence class Show Notations de

    Let SS be a set and RR be an equivalence relation on SS, then for any we call xShttp://mathhub.info/smglom/sets/set.omdoc?set?insetxS we call the set (fundefeq[xR][x]R)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqxRhttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?equivalence-classxR the equivalence class of xx (under RR), and the set (fundefeq[xR]S_R)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqxRhttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?quotient-spaceSR the quotient space of SS (under RR).

    The mapping (projectionR):S(S_R);x([x]R)http://mathhub.info/smglom/sets/functions.omdoc?functions?funsuchthathttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?projectionRShttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?quotient-spaceSRxhttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?equivalence-classxR is called the projection of SS to S_Rhttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?quotient-spaceSR.


  • equivalence relation de

    A relation R(A×A)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetRhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAA is an equivalence relation on AA, iff RR is reflexive, symmetric, and transitive.


  • Erdös-Borwein constant Show Notations de

    The Erdös-Borwein constant is the sum of the reciprocals of the Mersenne numbers.

    Ehttp://mathhub.info/smglom/smglom/erdoesborweinconstant.omdoc?erdoesborweinconstant?Erdoes-Borwein-constant

  • Euclid number Show Notations de

  • Euler-Mascheroni constant

    The Euler-Mascheroni constant (also called Euler’s constant) is a mathematical constant. It is defined as the limiting difference between the harmonic series and the natural logarithm:

    γhttp://mathhub.info/smglom/smglom/eulermascheroni.omdoc?eulermascheroni?eulermascheroni
    γ=0.5772156649015329...http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?startswithhttp://mathhub.info/smglom/smglom/eulermascheroni.omdoc?eulermascheroni?eulermascheroni

  • Euler’s constant Show Notations de

    The number ee, sometimes called Euler’s constant (also known as Napier’s constant) is an important mathematical constant.

    ehttp://mathhub.info/smglom/numberfields/eulersnumber.omdoc?eulersnumber?eulersnumber
    e=2.718281828459045...http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?startswithhttp://mathhub.info/smglom/numberfields/eulersnumber.omdoc?eulersnumber?eulersnumber

  • Euler’s constant

    The Euler-Mascheroni constant (also called Euler’s constant) is a mathematical constant. It is defined as the limiting difference between the harmonic series and the natural logarithm:

    γhttp://mathhub.info/smglom/smglom/eulermascheroni.omdoc?eulermascheroni?eulermascheroni
    γ=0.5772156649015329...http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?startswithhttp://mathhub.info/smglom/smglom/eulermascheroni.omdoc?eulermascheroni?eulermascheroni

  • Euler’s notation de

    If f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN is differentiable on MM, then the derivative function (Dxf):MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfxMN of ff (with respect to xx) is defined as

    (fundefeq[fxa]Dxf(a))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfxahttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa

    The dependency of Dxfhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfx on xx is left implicit in this notation (Lagrange’s notation). In Leibniz’s notation we write the derivative of ff with respect to xx as Dxfhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfx and the derivative of ff at aa as Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa or Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa. In Euler’s notation, this is written as Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa and in Newton’snotation as Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa.


  • factorial Show Notations de

    The factorial of a non-negative integer nn, denoted by nfactorialhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnhttp://mathhub.info/smglom/numberfields/factorial.omdoc?factorial?factorial, is the product of all positive integers less than or equal to nn.


  • factorion de

    A factorion is a natural number that equals the sum of the factorial of its decimal digits.

    For example

    (multi-relation-expression145equal1+24+120equal(1!)+(4!)+(5!))http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/factorial.omdoc?factorial?factorialhttp://mathhub.info/smglom/numberfields/factorial.omdoc?factorial?factorialhttp://mathhub.info/smglom/numberfields/factorial.omdoc?factorial?factorial

  • Farey sequence de

    The Farey sequence of order nn is the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to nn, arranged in order of increasing size.

    Each Farey sequence starts with the value 0, denoted by the fraction 01http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?division, and ends with the value 1, denoted by the fraction 11http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?division (although some authors omit these terms).

    The Farey sequences of orders 1 to 4 are:

    (F1)=({(01),(11)})http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/smglom/fareysequence.omdoc?fareysequence?fareyseqhttp://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?division

    (F2)=({(01),(12),(11)})http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/smglom/fareysequence.omdoc?fareysequence?fareyseqhttp://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?division

    (F3)=({(01),(13),(12),(23),(11)})http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/smglom/fareysequence.omdoc?fareysequence?fareyseqhttp://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?division

    (F4)=({(01),(14),(13),(12),(23),(34),(11)})http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/smglom/fareysequence.omdoc?fareysequence?fareyseqhttp://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?division


  • Fermat number Show Notations de

    A Fermat number is a natural number of the form (fundefeq[n]Fn)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqnhttp://mathhub.info/smglom/numbers/fermatnumber.omdoc?fermatnumber?Fermat-numbern.


  • Fermat prime Show Notations de

    A Fermat prime is a prime number of the form

    (Fn)=((2(2n))+1)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/primes/fermatprime.omdoc?fermatprime?Fermat-primenhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn

  • Feynman point de

    The Feynman point is a sequence of six 9s that begins at the 762nd decimal place of the decimal representation of πhttp://mathhub.info/smglom/numberfields/pinumber.omdoc?pinumber?pinumber.


  • Fibonacci numbers Show Notations de

    The Fibonacci numbers or Fibonacci series or Fibonacci sequence are the numbers in the following integer sequence:

    0,1,1,2,3,5,8,13,21,34,55,89,144,...http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?infseq

    The first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the previous two.

    (Fn)=(defined-piecewise(
    0wenn(n=0)
    )
    (
    1wenn(n=1)
    )
    (
    ((F(n-1))+(F(n-2)))sonst
    )
    )
    http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/fibonaccinumbers.omdoc?fibonaccinumbers?Fibonacci-numbersnhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numbers/fibonaccinumbers.omdoc?fibonaccinumbers?Fibonacci-numbershttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numbers/fibonaccinumbers.omdoc?fibonaccinumbers?Fibonacci-numbershttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn

  • Fibonacci polynomials de

    The Fibonacci polynomials are defined by the recurrence relation

    ((Fn(x))x)=(defined-piecewise(
    0wenn(n=0)
    )
    (
    1wenn(n=1)
    )
    (
    ((x(F(n-1)(x))x)+((F(n-2)(x))x))wenn(n2)
    )
    )
    http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/fibonaccipolynomials.omdoc?fibonaccipolynomials?fibpolnxhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationxhttp://mathhub.info/smglom/smglom/fibonaccipolynomials.omdoc?fibonaccipolynomials?fibpolhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/fibonaccipolynomials.omdoc?fibonaccipolynomials?fibpolhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnxhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methann

    The first few Fibonacci polynomials are:

    F0(x)http://mathhub.info/smglom/smglom/fibonaccipolynomials.omdoc?fibonaccipolynomials?fibpol

    ((F1(x))x)=1http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/fibonaccipolynomials.omdoc?fibonaccipolynomials?fibpolx

    ((F2(x))x)=xhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/fibonaccipolynomials.omdoc?fibonaccipolynomials?fibpolxx

    ((F3(x))x)=((x2)+1)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/fibonaccipolynomials.omdoc?fibonaccipolynomials?fibpolxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationx

    ((F4(x))x)=((x3)+(2x))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/fibonaccipolynomials.omdoc?fibonaccipolynomials?fibpolxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationx

    ((F5(x))x)=((x4)+(3(x2))+1)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/fibonaccipolynomials.omdoc?fibonaccipolynomials?fibpolxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationx


  • Fibonacci sequence Show Notations de

    The Fibonacci numbers or Fibonacci series or Fibonacci sequence are the numbers in the following integer sequence:

    0,1,1,2,3,5,8,13,21,34,55,89,144,...http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?infseq

    The first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the previous two.

    (Fn)=(defined-piecewise(
    0wenn(n=0)
    )
    (
    1wenn(n=1)
    )
    (
    ((F(n-1))+(F(n-2)))sonst
    )
    )
    http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/fibonaccinumbers.omdoc?fibonaccinumbers?Fibonacci-numbersnhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numbers/fibonaccinumbers.omdoc?fibonaccinumbers?Fibonacci-numbershttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numbers/fibonaccinumbers.omdoc?fibonaccinumbers?Fibonacci-numbershttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn

  • Fibonacci series Show Notations de

    The Fibonacci numbers or Fibonacci series or Fibonacci sequence are the numbers in the following integer sequence:

    0,1,1,2,3,5,8,13,21,34,55,89,144,...http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?infseq

    The first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the previous two.

    (Fn)=(defined-piecewise(
    0wenn(n=0)
    )
    (
    1wenn(n=1)
    )
    (
    ((F(n-1))+(F(n-2)))sonst
    )
    )
    http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/fibonaccinumbers.omdoc?fibonaccinumbers?Fibonacci-numbersnhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numbers/fibonaccinumbers.omdoc?fibonaccinumbers?Fibonacci-numbershttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numbers/fibonaccinumbers.omdoc?fibonaccinumbers?Fibonacci-numbershttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn

  • field de

    A field is a ring with multiplicative inverses.


  • fifteenth smarandache constant de

    The fifteenth smarandache constant http://mathhub.info/smglom/smglom/smarandacheconstant15.omdoc?smarandacheconstant15?fifteenth-smarandache-constant is defind by

    http://mathhub.info/smglom/smglom/smarandacheconstant15.omdoc?smarandacheconstant15?fifteenth-smarandache-constant

    where S(n)http://mathhub.info/smglom/smglom/smarandachefunction.omdoc?smarandachefunction?smarandache-functionn is the smarandache function.

    The sum converges.


  • fifth smarandache constant Show Notations de

    The fifth smarandache constant http://mathhub.info/smglom/smglom/smarandacheconstant5.omdoc?smarandacheconstant5?fifth-smarandache-constant is defind by

    http://mathhub.info/smglom/smglom/smarandacheconstant5.omdoc?smarandacheconstant5?fifth-smarandache-constant

    where S(n)http://mathhub.info/smglom/smglom/smarandachefunction.omdoc?smarandachefunction?smarandache-functionn is the smarandache function.

    The sum converges.


  • finite de

    We say that a set AA is finite and has cardinality (or size) (#(A))http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityAhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, iff there is a bijective function f:A({n|(n<(#(A)))})http://mathhub.info/smglom/sets/functions.omdoc?functions?funfAhttp://mathhub.info/smglom/sets/set.omdoc?set?rsetsthttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-numbernhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthannhttp://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityA.

    The cardinality of a set AA is also written as #(A)http://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityA, #(A)http://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityA, #(A)http://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityA, or #(A)http://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityA.


  • first Chebyshev function de

    The first Chebyshev function ϑ(x)http://mathhub.info/smglom/smglom/chebyshevfunction.omdoc?chebyshevfunction?firstchebyfuncx or θxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationθx is given by

    ϑ(x)http://mathhub.info/smglom/smglom/chebyshevfunction.omdoc?chebyshevfunction?firstchebyfuncx

    with the sum extending over all primes pp that are less than or equal to xx.


  • first derivative de

    For any nhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number we define the nnth derivative of a function f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN as

    (fundefeq[nfxa]dnfdxn|x=a)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqnfxahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa

    The first derivative d1dx1fhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtfxof ff is Dxfhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfx, d2dx2fhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtfx is the second derivative of ff, d3dx3fhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtfx the third derivative of ff, etc. In Leibniz’ notation the nnth derivative function of ff is denoted by dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx.

    We call ff nn times differentiable at aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM, iff dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa exists as a limit. Analogously, ff is nn times differentiable on MM, iff dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx exists and is total on MM.

    ff is called infinitely differentiable or smooth at aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM or on MM, iff dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa and dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx exist for all nhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number respectively.


  • first Skewes number Show Notations de

    The second Skewes number http://mathhub.info/smglom/smglom/skewesnumber.omdoc?skewesnumber?second-Skewes-number is the number above which (π(n))(Li(n))http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?NumberPrimeNumbernhttp://mathhub.info/smglom/smglom/logarithmicintegralbig.omdoc?logarithmicintegralbig?logarithmicintbign must fail assuming that the Riemann hypothesis is false.

    =(10(10(101000)))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/smglom/skewesnumber.omdoc?skewesnumber?second-Skewes-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiation

  • first smarandache constant Show Notations

    The first smarandache constant http://mathhub.info/smglom/smglom/smarandacheconstant1.omdoc?smarandacheconstant1?first-smarandache-constant is defind by

    http://mathhub.info/smglom/smglom/smarandacheconstant1.omdoc?smarandacheconstant1?first-smarandache-constant

    where S(n)http://mathhub.info/smglom/smglom/smarandachefunction.omdoc?smarandachefunction?smarandache-functionn is the smarandache function.


  • floor Show Notations de

    The floor

    r
    http://mathhub.info/smglom/numberfields/ceilingfloor.omdoc?ceilingfloor?floorr of a real number rr is the largest integer that is smaller or equal to rr. The floor function is also called the greatest integer function.


  • Fortunate number de

    A Fortunate number, named after Reo Fortune, for a given positive integer nn is the smallest integer m>1http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanm such that ((pn)#)+mhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/primorial.omdoc?primorial?primorialn%23m is a prime number, where the primorial (pn)#http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/primorial.omdoc?primorial?primorialn%23 is the product of the first nn prime numbers.

    A Fortunate prime is a Fortunate number which is also a prime number.


  • Fortunate prime

    A Fortunate number, named after Reo Fortune, for a given positive integer nn is the smallest integer m>1http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanm such that ((pn)#)+mhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/primorial.omdoc?primorial?primorialn%23m is a prime number, where the primorial (pn)#http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/primorial.omdoc?primorial?primorialn%23 is the product of the first nn prime numbers.

    A Fortunate prime is a Fortunate number which is also a prime number.


  • fourteenth smarandache constant

    The fourteenth smarandache constant fourteenth-smarandache-constanthttp://mathhub.info/smglom/smglom/smarandacheconstant14.omdoc?smarandacheconstant14?fourteenth-smarandache-constant is defind by

    s14(α)http://mathhub.info/smglom/smglom/smarandacheconstant14.omdoc?smarandacheconstant14?fourteenth-smarandache-constantα

    where S(n)http://mathhub.info/smglom/smglom/smarandachefunction.omdoc?smarandachefunction?smarandache-functionn is the smarandache function.

    The sum converges for all real numbers α>1http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanα.


  • Franel number Show Notations de

    The Franel numbers Frnhttp://mathhub.info/smglom/numbers/franelnumber.omdoc?franelnumber?Franel-numbern are the numbers

    Frnhttp://mathhub.info/smglom/numbers/franelnumber.omdoc?franelnumber?Franel-numbern

  • Franel number de

    The Franel numbers Frnhttp://mathhub.info/smglom/numbers/franelnumberrecurrence.omdoc?franelnumberrecurrence?franelnumberrecurrencen are a sequence of integers defind by the recurrence equation

    (Frn)=((((((7(n2))-(7n))+2)(Fr(n-1)))+(8((n-1)2)(Fr(n-2))))(n2))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/franelnumberrecurrence.omdoc?franelnumberrecurrence?franelnumberrecurrencenhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnhttp://mathhub.info/smglom/numbers/franelnumberrecurrence.omdoc?franelnumberrecurrence?franelnumberrecurrencehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numbers/franelnumberrecurrence.omdoc?franelnumberrecurrence?franelnumberrecurrencehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn

    where (Fr0)=1http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/franelnumberrecurrence.omdoc?franelnumberrecurrence?franelnumberrecurrence and (Fr1)=2http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/franelnumberrecurrence.omdoc?franelnumberrecurrence?franelnumberrecurrence.


  • friendly -tuple de

    nn numbers with the same abundancy form a friendly nn-tuple.


  • function de

    If we do not want to specify whether a partial function is total, then we simply speak of a function.


  • function space Show Notations de

    Given sets AA and BB we will call the set ABhttp://mathhub.info/smglom/sets/functions.omdoc?functions?function-spaceAB (ABhttp://mathhub.info/smglom/sets/functions.omdoc?functions?partial-function-spaceAB) of all (partial) functions from AA to BB the (partial) function space from AA to BB.


  • Gelfond-Schneider constant de

    The Gelfond-Schneider constant or Hilbert number is the following real number:

    (2(2))=(2.665144142690225)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?square-roothttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplication

  • Gelfond’s constant de

    Gelfond’s constant is the following real number

    (eπ)=23.14069263277927...http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?startswithhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/eulersnumber.omdoc?eulersnumber?eulersnumberhttp://mathhub.info/smglom/numberfields/pinumber.omdoc?pinumber?pinumber

  • general harmonic series de

    The general harmonic series are series of the form

    (infinite-sum0[n]1((an)+b))http://mathhub.info/smglom/calculus/infinitesum.omdoc?infinitesum?infinite-sumnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationanb

    where a0http://mathhub.info/smglom/mv/equal.omdoc?equal?notequala and bb are real numbers.


  • generalized Cullen number de

    A generalized Cullen number is a number of the form

    (nbn)+1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationbn

    where (n+2)>bhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnb.

    If a prime can be written in this form, it is then called a generalized Cullen prime.


  • generalized Cullen prime de

    A generalized Cullen number is a number of the form

    (nbn)+1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationbn

    where (n+2)>bhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnb.

    If a prime can be written in this form, it is then called a generalized Cullen prime.


  • generalized Woodall number de

  • geometric mean Show Notations de

    The geometric mean of a set M=({(a1,,an)})http://mathhub.info/smglom/mv/equal.omdoc?equal?equalMhttp://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlian is given by:

    (multi-relation-expressionGMequal(i=1nai)(1n)equal(a1a2an)n)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/geometricmean.omdoc?geometricmean?geometric-meanMhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdfromtoniOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionnhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?rootnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6ef

  • girth Show Notations de

    For a graph GG, the minimum length of a cycle contained in GG is the girth g(G)http://mathhub.info/smglom/graphs/graphcycleparameters.omdoc?graphcycleparameters?girthG of GG. The maximum length of a cycle contained in GG is the circumference of GG. For a graph which does not contain a cycle the girth is set to (g(G))=http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/graphs/graphcycleparameters.omdoc?graphcycleparameters?girthGhttp://mathhub.info/smglom/numberfields/infinity.omdoc?infinity?infinity, its circumference is set to zero.


  • gleichmäïg konvergent

    Let AA be a set, B,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureBd a metric space, and (sequenceon[f]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonfn a sequence of functions (fi):ABhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funhttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselfiAB, then we call (sequenceon[f]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonfn uniformly convergent to f:ABhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfAB on AAhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efA, iff for every ϵ0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methanϵ, there exists a Nhttp://mathhub.info/smglom/sets/set.omdoc?set?insetNhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, such that for all xAhttp://mathhub.info/smglom/sets/set.omdoc?set?insetxOMFOREIGNscala.xml.Node$@6040b6ef and all nNhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methannN we have (d)<ϵhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthandϵ.

    Ist AA eine Menge, B,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureBd ein metrischer Raum und (sequenceon[f]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonfn eine Folge von Funktionen (fi):ABhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funhttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselfiAB, dann nennen wir (sequenceon[f]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonfn gleichmäïg konvergent gegen f:ABhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfAB auf AAhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efA, falls zu jedem ϵ0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methanϵ ein Nhttp://mathhub.info/smglom/sets/set.omdoc?set?insetNhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number existiert, so dass (d)<ϵhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthandϵ für alle xAhttp://mathhub.info/smglom/sets/set.omdoc?set?insetxOMFOREIGNscala.xml.Node$@6040b6ef und alle nNhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methannN.


  • good de

    A binomial coefficient 𝒞knhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficientnk with k2http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methank is called good if its least prime factor (leastprimefactorhttp://mathhub.info/smglom/smglom/leastprimefactor.omdoc?leastprimefactor?leastprimefactor) satisfies

    (leastprimefactor(𝒞kn))>khttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/leastprimefactor.omdoc?leastprimefactor?leastprimefactorhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficientnkk

  • good prime de

  • googol de

    A googol is the large number 10100http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiation.


  • googolminex de

    A googolminex is the reciprocal of the googolplex.


  • googolplex de

    A googolplexplexplex is the number

    (googolplexplexplex)=(10(googolplexplex))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationgoogolplexplexplexhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationgoogolplexplex

    A googolplex is the number

    (multi-relation-expressiongoogolplexequal10(googol)equal10(10100))http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationgoogolplexhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationgoogolhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiation

    A googolplexplex is the number

    (googolplexplex)=(10(googolplex))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationgoogolplexplexhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationgoogolplex

  • googolplexian de

    A googolplexian is the number

    (multi-relation-expressiongoogolplexianequalgoogolplexplexequal10(googolplex))http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationgoogolplexianhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationgoogolplexplexhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationgoogolplex

  • googolplexplex

    A googolplexplexplex is the number

    (googolplexplexplex)=(10(googolplexplex))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationgoogolplexplexplexhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationgoogolplexplex

    A googolplexplex is the number

    (googolplexplex)=(10(googolplex))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationgoogolplexplexhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationgoogolplex

  • googolplexplexplex

    A googolplexplexplex is the number

    (googolplexplexplex)=(10(googolplexplex))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationgoogolplexplexplexhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationgoogolplexplex

  • graph de

    A graph is a pair V,Ehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE such that VV is a set and E(V×V)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetEhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsVV is a subset of the set of pairs from VV. We call VV the vertices (or nodes, points,junctions) and EE the edges (or lines, branches, arcs) of GG.


  • graph de

    The graph of a function ff is the set of all pairs x,(fx)http://mathhub.info/smglom/sets/pair.omdoc?pair?pairxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationfx.


  • greatest common divisor de

    The greatest common divisor (gcd), also known as the greatest common factor (gcf), or highest common factor (hcf), of two or more integers (at least one of which is not zero), is the largest positive integer that divides the numbers without a remainder.


  • greatest common factor de

    The greatest common divisor (gcd), also known as the greatest common factor (gcf), or highest common factor (hcf), of two or more integers (at least one of which is not zero), is the largest positive integer that divides the numbers without a remainder.


  • greatest integer function Show Notations de

    The floor

    r
    http://mathhub.info/smglom/numberfields/ceilingfloor.omdoc?ceilingfloor?floorr of a real number rr is the largest integer that is smaller or equal to rr. The floor function is also called the greatest integer function.


  • greatest lower bound Show Notations de

    Let S,polehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureShttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole be an ordered set and TShttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetTS, then we call the smallest upper bound sup(T)http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?supremumT (largest lower bound inf(T)http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?infimumT) of TT the supremum or least upper bound (infimum or greatest lower bound) of TT (if it exists).

    If ee is an expression and φφ a condition (in a variable xx), we write (bsupremumφ[x]e)http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?bsupremumφxe for sup((bsetst[x]e))http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?supremumhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstxe and call it the supremum for ee over φφ. Analogously, we write (binfimumφ[x]e)http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?binfimumφxe for inf((bsetst[x]φ))http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?infimumhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstxφ and call it the infimum for ee over φφ


  • Gregory number de

  • group de

    A group is a magma Ghttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureG, such that (aa)=(bb)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionaahttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionbb, (a(bb))=ahttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionahttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionbba, ((aa)(bc))=(cb)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionaahttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionbchttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisioncb, and ((ac)(bc))=(ab)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionachttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionbchttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionab for all a,b,cGhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabcG.


  • happy prime

    A happy prime is a number that is both happy and prime.


  • Hardy-Ramanujan number de

    The Hardy-Ramanujan number 1729 is the smallest number expressible as the sum of two cubes in two different ways.

    (multi-relation-expression1729equal(13)+(123)equal(93)+(103))http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiation

  • harmonic divisor number de

    A harmonic divisor number, or Ore number, is a positive integer whose divisors have a harmonic mean that is an integer.


  • harmonic mean Show Notations de

  • harmonic number

    The nn-th harmonic number is the sum of the reciprocals of the first nn natural numbers:

    Hnhttp://mathhub.info/smglom/numbers/harmonicnumber.omdoc?harmonicnumber?harmonicnumbern

  • harmonic series de

    The harmonic series is the divergent infinite series:

    (infinite-sum1[n]1n)http://mathhub.info/smglom/calculus/infinitesum.omdoc?infinitesum?infinite-sumnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionn

  • Harshad number de

    A Harshad number or Niven number in a given number base is an integer that is divisible by the sum of its digits when written in that base.


  • Hausdorff limit Show Notations de

    Let X,Ohttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureXOMFOREIGNscala.xml.Node$@6040b6ef be a Hausdorff space, then we say that a sequence (sequenceon[name.cvar.0]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonname.cvar.0n converges to xXhttp://mathhub.info/smglom/sets/set.omdoc?set?insetxX (we call xx the Hausdorff limit and write xx), iff every neighborhood of xx only contains finitely many xiOMFOREIGNscala.xml.Node$@6040b6ef.


  • Hausdorff space

    Let X,Ohttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureXOMFOREIGNscala.xml.Node$@6040b6ef be a topological space, then we call OOMFOREIGNscala.xml.Node$@6040b6ef a Hausdorff topology (and X,Ohttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureXOMFOREIGNscala.xml.Node$@6040b6ef a Hausdorff space), iff for all p,qXhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetpqX with pqhttp://mathhub.info/smglom/mv/equal.omdoc?equal?notequalpq, there are disjoint open sets P,QOhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetPQOMFOREIGNscala.xml.Node$@6040b6ef with pPhttp://mathhub.info/smglom/sets/set.omdoc?set?insetpP and qQhttp://mathhub.info/smglom/sets/set.omdoc?set?insetqQ.


  • Hausdorff topology

    Let X,Ohttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureXOMFOREIGNscala.xml.Node$@6040b6ef be a topological space, then we call OOMFOREIGNscala.xml.Node$@6040b6ef a Hausdorff topology (and X,Ohttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureXOMFOREIGNscala.xml.Node$@6040b6ef a Hausdorff space), iff for all p,qXhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetpqX with pqhttp://mathhub.info/smglom/mv/equal.omdoc?equal?notequalpq, there are disjoint open sets P,QOhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetPQOMFOREIGNscala.xml.Node$@6040b6ef with pPhttp://mathhub.info/smglom/sets/set.omdoc?set?insetpP and qQhttp://mathhub.info/smglom/sets/set.omdoc?set?insetqQ.


  • heteromecic number

    A pronic number Pnhttp://mathhub.info/smglom/numbers/pronicnumber.omdoc?pronicnumber?pronic-numbern, oblong number, rectangular number or heteromecic number, is a number which is the product of two consecutive natural numbers.

    (multi-relation-expressionPnequal(nn+1)equal(n2)+n)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numbers/pronicnumber.omdoc?pronicnumber?pronic-numbernhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationnn

  • highest common factor de

    The greatest common divisor (gcd), also known as the greatest common factor (gcf), or highest common factor (hcf), of two or more integers (at least one of which is not zero), is the largest positive integer that divides the numbers without a remainder.


  • highly composite number Show Notations de

    A highly composite number (HCN) is a positive integer with more divisors than any smaller positive integer.


  • Hilbert number de

    The Gelfond-Schneider constant or Hilbert number is the following real number:

    (2(2))=(2.665144142690225)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?square-roothttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplication

  • Hilbert prime de

    A Hilbert prime is a Hilbert number that is not divisible by a smaller Hilbert number (other than 1).


  • hyperbolic cosine integral de

    The hyperbolic cosine integral Chi(x)http://mathhub.info/smglom/smglom/hyperboliccosineintegral.omdoc?hyperboliccosineintegral?hyperboliccosineintx or Chi(x)http://mathhub.info/smglom/smglom/hyperboliccosineintegral.omdoc?hyperboliccosineintegral?hyperboliccosineintx is defined by:

    Chi(x)http://mathhub.info/smglom/smglom/hyperboliccosineintegral.omdoc?hyperboliccosineintegral?hyperboliccosineintx

  • hyperbolic sine integral de

    The hyperbolic sine integral Shi(x)http://mathhub.info/smglom/smglom/hyperbolicsineintegral.omdoc?hyperbolicsineintegral?hyperbolicsineintx or Shi(x)http://mathhub.info/smglom/smglom/hyperbolicsineintegral.omdoc?hyperbolicsineintegral?hyperbolicsineintx is defined by:

    Shi(x)http://mathhub.info/smglom/smglom/hyperbolicsineintegral.omdoc?hyperbolicsineintegral?hyperbolicsineintx

  • identity function Show Notations de

    For a set AA, the identity function (IdA):AAhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funhttp://mathhub.info/smglom/sets/identity-function.omdoc?identity-function?identity-functionAAA on AA maps any aAhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaA to itself.


  • identity of indiscernibles de

    Let MM be a set, then we call a function d:(M×M)http://mathhub.info/smglom/sets/functions.omdoc?functions?fundhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsMMhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number a distance function (or metric ) on MM, iff for all x,y,zMhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetxyzM the following three identities hold:

    1. 1.

      (d)=0http://mathhub.info/smglom/mv/equal.omdoc?equal?equald iff x=yhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalxy (identity of indiscernibles),

    2. 2.

      (d)=(d)http://mathhub.info/smglom/mv/equal.omdoc?equal?equaldd (symmetry), and

    3. 3.

      (d)((d)+(d))http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethandhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additiondd (triangle inequality).

    We call M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd a metric space with base set MM and metric dd.


  • idoneal number

    A positive integer nn is an idoneal number if and only if it cannot be written as (ab)+(bc)+(ac)http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationabhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationbchttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationac for distinct positive integer aa, bb, and cc.


  • image Show Notations de

    Let f:ABhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfAB be a function, AAhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efA, and BBhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efB, then we call

    • (fundefeq[fname.cvar.2]f(A))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfname.cvar.2http://mathhub.info/smglom/sets/image.omdoc?image?imageoffOMFOREIGNscala.xml.Node$@6040b6ef the image of AOMFOREIGNscala.xml.Node$@6040b6ef under ff,

    • (fundefeq[f]𝐈𝐦(f))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfhttp://mathhub.info/smglom/sets/image.omdoc?image?imagef the image of ff, and

    • (fundefeq[fname.cvar.6]f-1(B))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfname.cvar.6http://mathhub.info/smglom/sets/image.omdoc?image?pre-imagefOMFOREIGNscala.xml.Node$@6040b6ef the pre-image of BOMFOREIGNscala.xml.Node$@6040b6ef under ff.


  • image Show Notations de

    Let f:ABhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfAB be a function, AAhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efA, and BBhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efB, then we call

    • (fundefeq[fname.cvar.2]f(A))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfname.cvar.2http://mathhub.info/smglom/sets/image.omdoc?image?imageoffOMFOREIGNscala.xml.Node$@6040b6ef the image of AOMFOREIGNscala.xml.Node$@6040b6ef under ff,

    • (fundefeq[f]𝐈𝐦(f))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfhttp://mathhub.info/smglom/sets/image.omdoc?image?imagef the image of ff, and

    • (fundefeq[fname.cvar.6]f-1(B))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfname.cvar.6http://mathhub.info/smglom/sets/image.omdoc?image?pre-imagefOMFOREIGNscala.xml.Node$@6040b6ef the pre-image of BOMFOREIGNscala.xml.Node$@6040b6ef under ff.


  • imaginary unit Show Notations de

    The set http://mathhub.info/smglom/numberfields/complexnumbers.omdoc?complexnumbers?complex-number of complex numbers contains numbers of the form a+(b𝑖)http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionahttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationbhttp://mathhub.info/smglom/numberfields/complexnumbers.omdoc?complexnumbers?imaginary-unit, where a,bhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number. We call 𝑖http://mathhub.info/smglom/numberfields/complexnumbers.omdoc?complexnumbers?imaginary-unit the imaginary unit.


  • incidence geometry

    We call a triple P,L,Ihttp://mathhub.info/smglom/mv/structure.omdoc?structure?structurePLI an incidence structure (or incidence geometry), with points PP, lines LL, and incidence realation I(P×L)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetIhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsPL (we say pPhttp://mathhub.info/smglom/sets/set.omdoc?set?insetpP is on ll, if (p,l)Ihttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairplI), iff

    • There are at least two points in PP.

    • For any two points p,qPhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetpqP, there is exactly one line lLhttp://mathhub.info/smglom/sets/set.omdoc?set?insetlL such that (p,l)Ihttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairplI and (q,l)Ihttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairqlI.

    • For every line lLhttp://mathhub.info/smglom/sets/set.omdoc?set?insetlL there are at least two points on ll.

    • For every line lLhttp://mathhub.info/smglom/sets/set.omdoc?set?insetlL there is at least one point pp with plhttp://mathhub.info/smglom/sets/set.omdoc?set?ninsetpl.


  • incidence realation

    We call a triple P,L,Ihttp://mathhub.info/smglom/mv/structure.omdoc?structure?structurePLI an incidence structure (or incidence geometry), with points PP, lines LL, and incidence realation I(P×L)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetIhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsPL (we say pPhttp://mathhub.info/smglom/sets/set.omdoc?set?insetpP is on ll, if (p,l)Ihttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairplI), iff

    • There are at least two points in PP.

    • For any two points p,qPhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetpqP, there is exactly one line lLhttp://mathhub.info/smglom/sets/set.omdoc?set?insetlL such that (p,l)Ihttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairplI and (q,l)Ihttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairqlI.

    • For every line lLhttp://mathhub.info/smglom/sets/set.omdoc?set?insetlL there are at least two points on ll.

    • For every line lLhttp://mathhub.info/smglom/sets/set.omdoc?set?insetlL there is at least one point pp with plhttp://mathhub.info/smglom/sets/set.omdoc?set?ninsetpl.


  • incidence structure

    We call a triple P,L,Ihttp://mathhub.info/smglom/mv/structure.omdoc?structure?structurePLI an incidence structure (or incidence geometry), with points PP, lines LL, and incidence realation I(P×L)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetIhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsPL (we say pPhttp://mathhub.info/smglom/sets/set.omdoc?set?insetpP is on ll, if (p,l)Ihttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairplI), iff

    • There are at least two points in PP.

    • For any two points p,qPhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetpqP, there is exactly one line lLhttp://mathhub.info/smglom/sets/set.omdoc?set?insetlL such that (p,l)Ihttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairplI and (q,l)Ihttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairqlI.

    • For every line lLhttp://mathhub.info/smglom/sets/set.omdoc?set?insetlL there are at least two points on ll.

    • For every line lLhttp://mathhub.info/smglom/sets/set.omdoc?set?insetlL there is at least one point pp with plhttp://mathhub.info/smglom/sets/set.omdoc?set?ninsetpl.


  • indegree

    Let G=(V,E)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalGhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE be a directed graph and vVhttp://mathhub.info/smglom/sets/set.omdoc?set?insetvV a vertex in GG, then we define

    • indegree indeg(v)http://mathhub.info/smglom/graphs/inout-degree.omdoc?inout-degree?indegv of vv as #((bsetst[w]w))http://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstww

    • outdegree outdeg(v)http://mathhub.info/smglom/graphs/inout-degree.omdoc?inout-degree?outdegv of vv as #((bsetst[w]w))http://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstww


  • index of multiplication de

    We define the product over a sequence aiOMFOREIGNscala.xml.Node$@6040b6ef by

    (i=nmai):=(defined-piecewise(
    1wenn(n=m)
    )
    (
    (an)sonst
    )
    )
    http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdfromtonmiOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnmhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationOMFOREIGNscala.xml.Node$@6040b6ef

    The variable ii is called the index of multiplication and nn and mm the lower and upper bounds of the product respectively, together the specify the range of multiplication.

    There are variant product operators φaihttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdCollφOMFOREIGNscala.xml.Node$@6040b6ef and iSaihttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdInCollSiOMFOREIGNscala.xml.Node$@6040b6ef. The first one specifies the range of the product via a formula φφ in ii and the second one directly by giving a set SS.


  • index of summation de

    Summation is iterated addition, we define the sum over a sequnce aiOMFOREIGNscala.xml.Node$@6040b6ef by

    (i=nmai):=(defined-piecewise(
    0wenn(n=m)
    )
    (
    (an)sonst
    )
    )
    http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/numberfields/sum.omdoc?sum?sumnmiOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnmhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionOMFOREIGNscala.xml.Node$@6040b6ef

    The variable ii is called the index of summation and nn and mm the lower and upper bound of the sum respectively, together the specify the range of summation.

    There are variant summation operators (SumCollφ[i]ai)http://mathhub.info/smglom/numberfields/sum.omdoc?sum?SumCollφiOMFOREIGNscala.xml.Node$@6040b6ef and iSaihttp://mathhub.info/smglom/numberfields/sum.omdoc?sum?SumInCollSiOMFOREIGNscala.xml.Node$@6040b6ef. The first one specifies the range of the summation via a formula φφ in ii and the second one directly by giving a set SS.


  • induced cycle

    If in a graph GG which contains a cycle CC, an edge of GG joins two vertices of CC but is not itself an edge of the cycle, then this edge is a chord of the cycle CC. Thus a cycle is chordless iff it is an induced cycle in GG.


  • inferior limit Show Notations

    Ist M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd ein metrischer Raum und (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan eine Folge mit (ai)Mhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselaiM für ihttp://mathhub.info/smglom/sets/set.omdoc?set?insetihttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, so ist der

    • Limes superior limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (schreibe auch limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) definiert durch

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai
    • Limes inferior limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (schreibe auch limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) definiert durch

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai

    Let M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd be a metric space and (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan a sequence with (ai)Mhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselaiM for all ihttp://mathhub.info/smglom/sets/set.omdoc?set?insetihttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, then the

    • limit superior (also called supremum limit, superior limit, upper limit, or outer limit) limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (also written as limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) is defined as

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai
    • limit inferior (also called infimum limit, inferior limit, lower limit, or inner limit) limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (also written as limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) is defined as

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai

  • infimum Show Notations de

    Let S,polehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureShttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole be an ordered set and TShttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetTS, then we call the smallest upper bound sup(T)http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?supremumT (largest lower bound inf(T)http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?infimumT) of TT the supremum or least upper bound (infimum or greatest lower bound) of TT (if it exists).

    If ee is an expression and φφ a condition (in a variable xx), we write (bsupremumφ[x]e)http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?bsupremumφxe for sup((bsetst[x]e))http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?supremumhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstxe and call it the supremum for ee over φφ. Analogously, we write (binfimumφ[x]e)http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?binfimumφxe for inf((bsetst[x]φ))http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?infimumhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstxφ and call it the infimum for ee over φφ


  • infimum limit Show Notations

    Ist M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd ein metrischer Raum und (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan eine Folge mit (ai)Mhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselaiM für ihttp://mathhub.info/smglom/sets/set.omdoc?set?insetihttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, so ist der

    • Limes superior limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (schreibe auch limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) definiert durch

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai
    • Limes inferior limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (schreibe auch limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) definiert durch

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai

    Let M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd be a metric space and (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan a sequence with (ai)Mhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselaiM for all ihttp://mathhub.info/smglom/sets/set.omdoc?set?insetihttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, then the

    • limit superior (also called supremum limit, superior limit, upper limit, or outer limit) limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (also written as limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) is defined as

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai
    • limit inferior (also called infimum limit, inferior limit, lower limit, or inner limit) limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (also written as limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) is defined as

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai

  • infinite de

    A set that is not finite is called infinite.


  • infinite sum Show Notations de

    Let (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan be a sequence with (an)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselanhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number or (an)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselanhttp://mathhub.info/smglom/numberfields/complexnumbers.omdoc?complexnumbers?complex-number. the (formal) infinite sum (infinite-sum1[n]an)http://mathhub.info/smglom/calculus/infinitesum.omdoc?infinitesum?infinite-sumnhttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselan is defined as the limit limn(sn)http://mathhub.info/smglom/calculus/sequenceConvergence.omdoc?sequenceConvergence?limitnhttp://mathhub.info/smglom/calculus/series.omdoc?series?partial-suman of the sequence of partial sums, if it exists.


  • infinitely differentiable de

    We call ff nn times differentiable at aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM, iff dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa exists as a limit. Analogously, ff is nn times differentiable on MM, iff dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx exists and is total on MM.

    ff is called infinitely differentiable or smooth at aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM or on MM, iff dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa and dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx exist for all nhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number respectively.


  • Infinity Show Notations de

    Infinity (written http://mathhub.info/smglom/numberfields/infinity.omdoc?infinity?infinity) is an abstract concept describing something without any limit. In mathematics is is usually treated like a number.


  • injective de

    A function f:SThttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfST is called injective iff (f)=(f)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalff entails x=yhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalxy for all x,yShttp://mathhub.info/smglom/sets/set.omdoc?set?minsetxyS.


  • inner limit Show Notations

    Ist M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd ein metrischer Raum und (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan eine Folge mit (ai)Mhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselaiM für ihttp://mathhub.info/smglom/sets/set.omdoc?set?insetihttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, so ist der

    • Limes superior limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (schreibe auch limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) definiert durch

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai
    • Limes inferior limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (schreibe auch limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) definiert durch

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai

    Let M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd be a metric space and (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan a sequence with (ai)Mhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselaiM for all ihttp://mathhub.info/smglom/sets/set.omdoc?set?insetihttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, then the

    • limit superior (also called supremum limit, superior limit, upper limit, or outer limit) limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (also written as limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) is defined as

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai
    • limit inferior (also called infimum limit, inferior limit, lower limit, or inner limit) limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (also written as limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) is defined as

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai

  • integer de

    The set http://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?integers of integer numbers (or integers)is the set {(,(-(2)),(-(1)),0,1,2,)}http://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?dotsaseqdotshttp://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?uminushttp://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?uminus. Any member of http://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?integers is called an integer.

    The set negative integers of http://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?negative-integers is the set {(,(-(3)),(-(2)),(-(1)))}http://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?dotsaseqhttp://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?uminushttp://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?uminushttp://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?uminus.


  • integer division

    The integer division operator computes the integer quotient (or modulus) ndivmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divnm of two natural numbers. ndivmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divnm is defined as that qhttp://mathhub.info/smglom/sets/set.omdoc?set?insetqhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, such that n=((mq)+r)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationmqr for some 0r<mhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?betweenesrm. The number rr is called the remainder and is written as nmodmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?modnm.


  • integer interval Show Notations de

    We define the integer interval as a subset of consecutive integers: (fundefeq[ab][a,b])http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqabhttp://mathhub.info/smglom/calculus/interval.omdoc?interval?integer-intervalab


  • integer number Show Notations de

    The set http://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?integers of integer numbers (or integers)is the set {(,(-(2)),(-(1)),0,1,2,)}http://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?dotsaseqdotshttp://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?uminushttp://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?uminus. Any member of http://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?integers is called an integer.

    The set negative integers of http://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?negative-integers is the set {(,(-(3)),(-(2)),(-(1)))}http://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?dotsaseqhttp://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?uminushttp://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?uminushttp://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?uminus.


  • integer quotient

    The integer division operator computes the integer quotient (or modulus) ndivmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divnm of two natural numbers. ndivmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divnm is defined as that qhttp://mathhub.info/smglom/sets/set.omdoc?set?insetqhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, such that n=((mq)+r)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationmqr for some 0r<mhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?betweenesrm. The number rr is called the remainder and is written as nmodmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?modnm.


  • integers Show Notations de

    The set http://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?integers of integer numbers (or integers)is the set {(,(-(2)),(-(1)),0,1,2,)}http://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?dotsaseqdotshttp://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?uminushttp://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?uminus. Any member of http://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?integers is called an integer.

    The set negative integers of http://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?negative-integers is the set {(,(-(3)),(-(2)),(-(1)))}http://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?dotsaseqhttp://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?uminushttp://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?uminushttp://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?uminus.


  • interprime de

    An interprime is the arithmetic mean of two consecutive odd primes.


  • intersection Show Notations tr de ro

    Let II be a set and SiOMFOREIGNscala.xml.Node$@6040b6ef a family of sets indexed by II, then the intersection iISihttp://mathhub.info/smglom/sets/intersection.omdoc?intersection?mintersectCollectionIiOMFOREIGNscala.xml.Node$@6040b6ef over II is (bsetst[x]x)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstxx.


  • intersection Show Notations tr de ro

    Let AA and BB be sets, then the intersection ABhttp://mathhub.info/smglom/sets/intersection.omdoc?intersection?intersectionAB of AA and BB is (bsetst[x]x)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstxx.


  • inverse function Show Notations de

    If f:ABhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfAB is injective, then the converse relation is a partial function (f-1):BAhttp://mathhub.info/smglom/sets/functions.omdoc?functions?partfunhttp://mathhub.info/smglom/sets/inverse-function.omdoc?inverse-function?inverse-functionfBA, we call it the inverse function of ff. If ff is bijective total, then f-1http://mathhub.info/smglom/sets/inverse-function.omdoc?inverse-function?inverse-functionf is a total function.


  • irrational number de

  • irreflexive de

    A relation R(A×A)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetRhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAA is called

    • reflexive on AA, iff (a,a)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairaaR for all aAhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaA, and

    • irreflexive (or anti-reflexive) on AA, iff (a,a)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?ninsethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairaaR for all aAhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaA.


  • Jacobi-Madden equation

    The Jacobi-Madden equation is the Diophantine equation

    ((a4)+(b4)+(c4)+(d4))=((a+b+c+d)4)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationahttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationbhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationchttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationdhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionabcd

  • Jacobsthal numbers de

    The Jacobsthal numbers are defined by the recurrence relation

    J0http://mathhub.info/smglom/numbers/jacobsthalnumbers.omdoc?jacobsthalnumbers?jacobsthalnumbers equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal 0
    J1http://mathhub.info/smglom/numbers/jacobsthalnumbers.omdoc?jacobsthalnumbers?jacobsthalnumbers equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal 1
    Jnhttp://mathhub.info/smglom/numbers/jacobsthalnumbers.omdoc?jacobsthalnumbers?jacobsthalnumbersn equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal (J(n-1))+(2(J(n-2)))http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numbers/jacobsthalnumbers.omdoc?jacobsthalnumbers?jacobsthalnumbershttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numbers/jacobsthalnumbers.omdoc?jacobsthalnumbers?jacobsthalnumbershttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn

    The definition in closed form:

    (Jn)=(((2n)-((1)n))3)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/jacobsthalnumbers.omdoc?jacobsthalnumbers?jacobsthalnumbersnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn

  • Jacobsthal-Lucas numbers

    The Jacobsthal-Lucas numbers are defined by the recurrence relation

    (Ln)=(defined-piecewise(
    2wenn(n=0)
    )
    (
    1wenn(n=1)
    )
    (
    ((L(n-1))+(2(L(n-2))))sonst
    )
    )
    http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/jacobsthallucasnumbers.omdoc?jacobsthallucasnumbers?Jacobsthal-Lucas-numbernhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numbers/jacobsthallucasnumbers.omdoc?jacobsthallucasnumbers?Jacobsthal-Lucas-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numbers/jacobsthallucasnumbers.omdoc?jacobsthallucasnumbers?Jacobsthal-Lucas-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn

    The definition in closed form: (Ln)=((2n)+((1)n))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/jacobsthallucasnumbers.omdoc?jacobsthallucasnumbers?Jacobsthal-Lucas-numbernhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn


  • junction de

    A graph is a pair V,Ehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE such that VV is a set and E(V×V)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetEhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsVV is a subset of the set of pairs from VV. We call VV the vertices (or nodes, points,junctions) and EE the edges (or lines, branches, arcs) of GG.


  • Kempner series

    The Kempner series Kmhttp://mathhub.info/smglom/smglom/kempnerseries.omdoc?kempnerseries?kempnerseriesm are subharmonic series formed by omitting all terms from the harmonic series whose denominator expressed in base 10 contains a mm digit (or a digit-sequence like mm):

    Kmhttp://mathhub.info/smglom/smglom/kempnerseries.omdoc?kempnerseries?kempnerseriesm

    For example

    (K9)=(1+(12)+(13)+(14)++(17)+(18)+(110)++(117)+(118)+(120)++(187)+(188)+(1100)+((1101)))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/smglom/kempnerseries.omdoc?kempnerseries?kempnerserieshttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?division

    (K9)=22.92067...http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?startswithhttp://mathhub.info/smglom/smglom/kempnerseries.omdoc?kempnerseries?kempnerseries

    (K314159)=2302582.3338637827...http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?startswithhttp://mathhub.info/smglom/smglom/kempnerseries.omdoc?kempnerseries?kempnerseries


  • Knoedel number Show Notations de

    A Knoedel number for a given positive integer nn is a composite number mm with the property that each imhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanim coprime to mm satisfies ((i(m-n))mod1)mhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/congruence.omdoc?congruence?congruent-modulohttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationihttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionmnm. The set of all Knoedel numbers for nn is denoted by Knhttp://mathhub.info/smglom/numbers/knoedelnumber.omdoc?knoedelnumber?Knoedel-numbern.


  • Kronecker delta

    The Kronecker delta is a function of two integers. The function is 1 if the variables are equal, and 0 otherwise:

    (δij)=(defined-piecewise(
    0wenn(ij)
    )
    (
    1wenn(i=j)
    )
    )
    http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/smglom/kroneckerdelta.omdoc?kroneckerdelta?kroneckerdeltaijhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?notequalijhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalij

  • Kynea number de

    A Kynea number is an integer of the form

    (((4n)+(2(n+1)))-1)=((((2n)+1)2)-2)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn

    for a positive integer nn.


  • Lagrange’s notation de

    If f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN is differentiable on MM, then the derivative function (Dxf):MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfxMN of ff (with respect to xx) is defined as

    (fundefeq[fxa]Dxf(a))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfxahttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa

    The dependency of Dxfhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfx on xx is left implicit in this notation (Lagrange’s notation). In Leibniz’s notation we write the derivative of ff with respect to xx as Dxfhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfx and the derivative of ff at aa as Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa or Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa. In Euler’s notation, this is written as Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa and in Newton’snotation as Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa.


  • lazy caterer’s sequence de

    The central polygonal numbers or lazy caterer’s sequence describes the maximum number pp of pieces of a circle (or a plane) that can be made with a given number nn of straight cuts.

    (multi-relation-expressionpequal(𝒞2n)+(𝒞1n)+(𝒞0n)equal((n2)+n+2)2)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionphttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficientnhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficientnhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficientnhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationnn

  • leaf

    A tree is a directed acyclic graph G=(V,E)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalGhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE such that

    • there is exactly one initialnode vrVhttp://mathhub.info/smglom/sets/set.omdoc?set?insetOMFOREIGNscala.xml.Node$@6040b6efV (called the root), and

    • all nodes but the root have indegree 1.

    We call vv the parent of ww, iff (v,w)Ehttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairvwE (ww is a child of vv). We call a node vv a leaf of GG, iff it is terminal, i.e. if it does not have children.


  • least common multiple Show Notations de

    The least common multiple (lcm) (also called the lowest common multiple or smallest common multiple) of two or more integers is the smallest positive integer that is divisible by all given integers.


  • least integer function Show Notations de

    The floor

    r
    http://mathhub.info/smglom/numberfields/ceilingfloor.omdoc?ceilingfloor?floorr of a real number rr is the largest integer that is smaller or equal to rr. The floor function is also called the greatest integer function.

    The ceiling ]]r[[http://mathhub.info/smglom/numberfields/ceilingfloor.omdoc?ceilingfloor?ceilingr of a real number rr is the smallest integer that is greater or equal to rr. The ceiling function is also called the least integer function.


  • least prime factor

    A prime factor of a natural number nn is a least prime factor lpf(n)http://mathhub.info/smglom/smglom/leastprimefactor.omdoc?leastprimefactor?leastprimefactorn if all prime factors of nn are equal or greater than it.


  • least upper bound Show Notations de

    Let S,polehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureShttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole be an ordered set and TShttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetTS, then we call the smallest upper bound sup(T)http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?supremumT (largest lower bound inf(T)http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?infimumT) of TT the supremum or least upper bound (infimum or greatest lower bound) of TT (if it exists).

    If ee is an expression and φφ a condition (in a variable xx), we write (bsupremumφ[x]e)http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?bsupremumφxe for sup((bsetst[x]e))http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?supremumhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstxe and call it the supremum for ee over φφ. Analogously, we write (binfimumφ[x]e)http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?binfimumφxe for inf((bsetst[x]φ))http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?infimumhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstxφ and call it the infimum for ee over φφ


  • Lebesgue identity de

    For all numbers mm, nn, pp and qq the Lebesgue identity is given by

    (((m2)+(n2)+(p2)+(q2))2)=((((2mq)+(2np))2)+(((2nq)-(2mp))2)+((((m2)+(n2))-(p2)-(q2))2))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationqhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationmqhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnqhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationmphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationq

  • left coset Show Notations

    Given an element gg of the group Ghttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureG and one of its subgroups HH, we define the left coset (respectively the right coset) of HH with gg as (fundefeq[gH]gH)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqgHhttp://mathhub.info/smglom/algebra/coset.omdoc?coset?left-cosetgH (respectively (fundefeq[gH]Hg)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqgHhttp://mathhub.info/smglom/algebra/coset.omdoc?coset?right-cosetgH).


  • left divides de

    Let Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR be a ring and m0http://mathhub.info/smglom/mv/equal.omdoc?equal?notequalm, then we say that mm is a left divisor of nn (or mm left divides nn) and that nn is a left multiple of mm, if there is a kRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetkR such that (multiplication)=nhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationn.

    Analogously we say that mm is a right divisor of nn (or mm right divides nn) and that nn is a right multiple of mm, if there is a kRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetkR such that (multiplication)=nhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationn.

    If RR is commxutative, then left and right divisors coincide and we simply speak of divisor and multiple and write m|nhttp://mathhub.info/smglom/algebra/ring-divisor.omdoc?ring-divisor?divisormn.

    1, 1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction, nn, and nhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn are known as the trivial divisors of nn. A divisor of nn that is not a trivial divisor is known as a proper or non-trivial divisor.


  • left divisor de

    Let Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR be a ring and m0http://mathhub.info/smglom/mv/equal.omdoc?equal?notequalm, then we say that mm is a left divisor of nn (or mm left divides nn) and that nn is a left multiple of mm, if there is a kRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetkR such that (multiplication)=nhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationn.

    Analogously we say that mm is a right divisor of nn (or mm right divides nn) and that nn is a right multiple of mm, if there is a kRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetkR such that (multiplication)=nhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationn.

    If RR is commxutative, then left and right divisors coincide and we simply speak of divisor and multiple and write m|nhttp://mathhub.info/smglom/algebra/ring-divisor.omdoc?ring-divisor?divisormn.

    1, 1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction, nn, and nhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn are known as the trivial divisors of nn. A divisor of nn that is not a trivial divisor is known as a proper or non-trivial divisor.


  • left multiple de

    Let Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR be a ring and m0http://mathhub.info/smglom/mv/equal.omdoc?equal?notequalm, then we say that mm is a left divisor of nn (or mm left divides nn) and that nn is a left multiple of mm, if there is a kRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetkR such that (multiplication)=nhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationn.

    Analogously we say that mm is a right divisor of nn (or mm right divides nn) and that nn is a right multiple of mm, if there is a kRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetkR such that (multiplication)=nhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationn.

    If RR is commxutative, then left and right divisors coincide and we simply speak of divisor and multiple and write m|nhttp://mathhub.info/smglom/algebra/ring-divisor.omdoc?ring-divisor?divisormn.

    1, 1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction, nn, and nhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn are known as the trivial divisors of nn. A divisor of nn that is not a trivial divisor is known as a proper or non-trivial divisor.


  • left-truncatable prime de

    A left-truncatable prime is a prime number which, in a given base, contains no 0, and if the leading (“left”) digit is successively removed, then all resulting numbers are prime.

    For example: 937

    A two-sided prime is both left-truncatable and right-truncatable.

    For example: 3137

    A right-truncatable prime is a prime number which remains prime when the last (“right”) digit is successively removed.

    For example: 3793


  • leftsided limit Show Notations de

    Let f:SThttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfST with S,Thttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?msseteqSThttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number and ahttp://mathhub.info/smglom/sets/set.omdoc?set?insetahttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number.

    Then the leftsided limit of ff at aa (also: the limit of ff as xx approaches aa from below) is defined to be the lhttp://mathhub.info/smglom/sets/set.omdoc?set?insetlhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number, sucht that for every ϵ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanϵ there is a δ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanδ such that (|(((f))l)|)ϵhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/numberfields/absolutevalue.omdoc?absolutevalue?absolute-valuehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionflϵ whenever (multi-relation-expression0lethan(a)xlethanδ)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionaxhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanδ. The leftsided limit is written as limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?leftsided-limitaxf, limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?leftsided-limitaxf, or limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?leftsided-limitaxf.

    Analogously, the rightsided limit at aa (also: the limit of ff as xx approaches aa from above) is the lhttp://mathhub.info/smglom/sets/set.omdoc?set?insetlhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number, such that for every ϵ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanϵ there is a δ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanδ such that (|(((f))l)|)ϵhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/numberfields/absolutevalue.omdoc?absolutevalue?absolute-valuehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionflϵ whenever (multi-relation-expression0lethan(x)alethanδ)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionxahttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanδ. The rightsided limit is written as limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?rightsided-limitaxf, limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?rightsided-limitaxf, or limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?rightsided-limitaxf.


  • Leibniz’s notation de

    If f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN is differentiable on MM, then the derivative function (Dxf):MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfxMN of ff (with respect to xx) is defined as

    (fundefeq[fxa]Dxf(a))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfxahttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa

    The dependency of Dxfhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfx on xx is left implicit in this notation (Lagrange’s notation). In Leibniz’s notation we write the derivative of ff with respect to xx as Dxfhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfx and the derivative of ff at aa as Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa or Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa. In Euler’s notation, this is written as Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa and in Newton’snotation as Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa.


  • length de

    Given a directed graph G=(V,E)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalGhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE we call a vector (p=((v0,,vn)))(V(n+1))http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalphttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlivnhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?nCartSpaceVhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionn a path in GG iff (vi-1,vi)Ehttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6efE for all 1inhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?betweeneein, n>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethann.

    • v0OMFOREIGNscala.xml.Node$@6040b6ef is called the start of pp (write start(p)http://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pstartp )

    • vnOMFOREIGNscala.xml.Node$@6040b6ef is called the end of pp (write end(p)http://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pendp )

    • nn is called the length of pp (write len(p)http://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?plenp )


  • Leyland number de

    A Leyland number is a number of the form

    (xy)+(yx)http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationxyhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationyx

    where xx and yy are integers greater than 1.

    The first Leyland numbers are 8,17,32,54,57,100,145,177,320,368,512,...http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?infseq


  • Limes inferior Show Notations

    Ist M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd ein metrischer Raum und (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan eine Folge mit (ai)Mhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselaiM für ihttp://mathhub.info/smglom/sets/set.omdoc?set?insetihttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, so ist der

    • Limes superior limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (schreibe auch limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) definiert durch

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai
    • Limes inferior limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (schreibe auch limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) definiert durch

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai

    Let M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd be a metric space and (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan a sequence with (ai)Mhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselaiM for all ihttp://mathhub.info/smglom/sets/set.omdoc?set?insetihttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, then the

    • limit superior (also called supremum limit, superior limit, upper limit, or outer limit) limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (also written as limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) is defined as

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai
    • limit inferior (also called infimum limit, inferior limit, lower limit, or inner limit) limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (also written as limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) is defined as

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai

  • Limes superior Show Notations

    Ist M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd ein metrischer Raum und (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan eine Folge mit (ai)Mhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselaiM für ihttp://mathhub.info/smglom/sets/set.omdoc?set?insetihttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, so ist der

    • Limes superior limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (schreibe auch limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) definiert durch

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai
    • Limes inferior limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (schreibe auch limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) definiert durch

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai

    Let M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd be a metric space and (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan a sequence with (ai)Mhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselaiM for all ihttp://mathhub.info/smglom/sets/set.omdoc?set?insetihttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, then the

    • limit superior (also called supremum limit, superior limit, upper limit, or outer limit) limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (also written as limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) is defined as

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai
    • limit inferior (also called infimum limit, inferior limit, lower limit, or inner limit) limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (also written as limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) is defined as

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai

  • limit Show Notations de

    Let A,dAhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureAOMFOREIGNscala.xml.Node$@6040b6ef and B,dBhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureBOMFOREIGNscala.xml.Node$@6040b6ef be metric spaces with MAhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetMA and NBhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetNB and f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN, then we say that lMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetlM is the limit of ff as xx approaches a limit point pNhttp://mathhub.info/smglom/sets/set.omdoc?set?insetpN (written limxp(f)http://mathhub.info/smglom/calculus/functionlimit.omdoc?functionlimit?limitpxf), iff for every ϵ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanϵ, there exists a δ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanδ such that (dA)ϵhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanOMFOREIGNscala.xml.Node$@6040b6efϵ whenever (multi-relation-expression0lethan(dB)lethanδ)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanδ.


  • limit inferior Show Notations

    Ist M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd ein metrischer Raum und (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan eine Folge mit (ai)Mhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselaiM für ihttp://mathhub.info/smglom/sets/set.omdoc?set?insetihttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, so ist der

    • Limes superior limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (schreibe auch limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) definiert durch

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai
    • Limes inferior limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (schreibe auch limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) definiert durch

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai

    Let M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd be a metric space and (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan a sequence with (ai)Mhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselaiM for all ihttp://mathhub.info/smglom/sets/set.omdoc?set?insetihttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, then the

    • limit superior (also called supremum limit, superior limit, upper limit, or outer limit) limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (also written as limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) is defined as

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai
    • limit inferior (also called infimum limit, inferior limit, lower limit, or inner limit) limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (also written as limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) is defined as

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai

  • limit point de

    Let SS be a subset of a topological space XX. A point xXhttp://mathhub.info/smglom/sets/set.omdoc?set?insetxX is a limit point of SS if every neighborhood of xx contains at least one sShttp://mathhub.info/smglom/sets/set.omdoc?set?insetsS with xshttp://mathhub.info/smglom/mv/equal.omdoc?equal?notequalxs.


  • limit superior Show Notations

    Ist M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd ein metrischer Raum und (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan eine Folge mit (ai)Mhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselaiM für ihttp://mathhub.info/smglom/sets/set.omdoc?set?insetihttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, so ist der

    • Limes superior limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (schreibe auch limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) definiert durch

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai
    • Limes inferior limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (schreibe auch limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) definiert durch

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai

    Let M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd be a metric space and (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan a sequence with (ai)Mhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselaiM for all ihttp://mathhub.info/smglom/sets/set.omdoc?set?insetihttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, then the

    • limit superior (also called supremum limit, superior limit, upper limit, or outer limit) limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (also written as limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) is defined as

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai
    • limit inferior (also called infimum limit, inferior limit, lower limit, or inner limit) limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (also written as limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) is defined as

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai

  • line de

    A graph is a pair V,Ehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE such that VV is a set and E(V×V)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetEhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsVV is a subset of the set of pairs from VV. We call VV the vertices (or nodes, points,junctions) and EE the edges (or lines, branches, arcs) of GG.


  • line

    We call a triple P,L,Ihttp://mathhub.info/smglom/mv/structure.omdoc?structure?structurePLI an incidence structure (or incidence geometry), with points PP, lines LL, and incidence realation I(P×L)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetIhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsPL (we say pPhttp://mathhub.info/smglom/sets/set.omdoc?set?insetpP is on ll, if (p,l)Ihttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairplI), iff

    • There are at least two points in PP.

    • For any two points p,qPhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetpqP, there is exactly one line lLhttp://mathhub.info/smglom/sets/set.omdoc?set?insetlL such that (p,l)Ihttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairplI and (q,l)Ihttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairqlI.

    • For every line lLhttp://mathhub.info/smglom/sets/set.omdoc?set?insetlL there are at least two points on ll.

    • For every line lLhttp://mathhub.info/smglom/sets/set.omdoc?set?insetlL there is at least one point pp with plhttp://mathhub.info/smglom/sets/set.omdoc?set?ninsetpl.


  • line graph Show Notations de

    For a graph G=(V,E)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalGhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE we call the graph L(G)http://mathhub.info/smglom/graphs/graphlinegraph.omdoc?graphlinegraph?linegraphG on EE, which has as its vertices the edges of GG and in which x,yEhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetxyE are adjacent as vertices in LGhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationLG if and only if they are adjacent as edges in GG, the line graph of GG.


  • linear combination

    Let VV be a vectorspace over a field FF and ({(v1,,vn)})Vhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsethttp://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlivnV, then we call any sum of scalar multiples of the viOMFOREIGNscala.xml.Node$@6040b6ef a linear combination of these.


  • linear order de

    We call a partial ordering RR a total ordering (or simple order or linear order), iff abhttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?poleab or bahttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?poleba for all a,bAhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabA.

    We call a structure S,polehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureShttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole of a set SS and a total ordering polehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole an totally ordered set.


  • linearly dependent

    Let VV be a vectorspace, then we call a set LVhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetLV linearly independent (else linearly dependent), iff no vLhttp://mathhub.info/smglom/sets/set.omdoc?set?insetvL is a linear combination of finitely many vectors in LL.


  • linearly independent

    Let VV be a vectorspace, then we call a set LVhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetLV linearly independent (else linearly dependent), iff no vLhttp://mathhub.info/smglom/sets/set.omdoc?set?insetvL is a linear combination of finitely many vectors in LL.


  • Liouville constant de

    The Liouville constant is a Liouville number defind by

    (infinite-sum1[j]10((j!)))http://mathhub.info/smglom/calculus/infinitesum.omdoc?infinitesum?infinite-sumjhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/factorial.omdoc?factorial?factorialj

  • Liouville number de

  • Lobb number Show Notations de

    The Lobb number Lm,nhttp://mathhub.info/smglom/numbers/lobbnumbers.omdoc?lobbnumbers?Lobb-numbermn counts the number of ways that n+mhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnm open parentheses and n-mhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnm close parentheses can be arranged to form the start of a valid sequence of balanced parentheses.

    (Lm,n)=((((2m)+1)(m+n+1))(𝒞(m+n)(2n)))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/lobbnumbers.omdoc?lobbnumbers?Lobb-numbermnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionmnhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficienthttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionmn

    where mm and nn are two integers with (multi-relation-expressionnmethanmmethan0)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionnhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methanmhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methan.


  • logarithmic integral de

  • lower de

    We define the product over a sequence aiOMFOREIGNscala.xml.Node$@6040b6ef by

    (i=nmai):=(defined-piecewise(
    1wenn(n=m)
    )
    (
    (an)sonst
    )
    )
    http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdfromtonmiOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnmhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationOMFOREIGNscala.xml.Node$@6040b6ef

    The variable ii is called the index of multiplication and nn and mm the lower and upper bounds of the product respectively, together the specify the range of multiplication.

    There are variant product operators φaihttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdCollφOMFOREIGNscala.xml.Node$@6040b6ef and iSaihttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdInCollSiOMFOREIGNscala.xml.Node$@6040b6ef. The first one specifies the range of the product via a formula φφ in ii and the second one directly by giving a set SS.


  • lower de

    Summation is iterated addition, we define the sum over a sequnce aiOMFOREIGNscala.xml.Node$@6040b6ef by

    (i=nmai):=(defined-piecewise(
    0wenn(n=m)
    )
    (
    (an)sonst
    )
    )
    http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/numberfields/sum.omdoc?sum?sumnmiOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnmhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionOMFOREIGNscala.xml.Node$@6040b6ef

    The variable ii is called the index of summation and nn and mm the lower and upper bound of the sum respectively, together the specify the range of summation.

    There are variant summation operators (SumCollφ[i]ai)http://mathhub.info/smglom/numberfields/sum.omdoc?sum?SumCollφiOMFOREIGNscala.xml.Node$@6040b6ef and iSaihttp://mathhub.info/smglom/numberfields/sum.omdoc?sum?SumInCollSiOMFOREIGNscala.xml.Node$@6040b6ef. The first one specifies the range of the summation via a formula φφ in ii and the second one directly by giving a set SS.


  • lower bound

    Let Shttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureS be a proset and TShttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetTS, then we call bShttp://mathhub.info/smglom/sets/set.omdoc?set?insetbS an upper bound of TT, iff (lessthan)http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthan for all tThttp://mathhub.info/smglom/sets/set.omdoc?set?insettT and an lower bound, iff (lessthan)http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthan for all tThttp://mathhub.info/smglom/sets/set.omdoc?set?insettT.


  • lower limit Show Notations

    Ist M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd ein metrischer Raum und (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan eine Folge mit (ai)Mhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselaiM für ihttp://mathhub.info/smglom/sets/set.omdoc?set?insetihttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, so ist der

    • Limes superior limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (schreibe auch limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) definiert durch

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai
    • Limes inferior limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (schreibe auch limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) definiert durch

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai

    Let M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd be a metric space and (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan a sequence with (ai)Mhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselaiM for all ihttp://mathhub.info/smglom/sets/set.omdoc?set?insetihttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, then the

    • limit superior (also called supremum limit, superior limit, upper limit, or outer limit) limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (also written as limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) is defined as

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai
    • limit inferior (also called infimum limit, inferior limit, lower limit, or inner limit) limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (also written as limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) is defined as

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai

  • lowest common multiple Show Notations de

    The least common multiple (lcm) (also called the lowest common multiple or smallest common multiple) of two or more integers is the smallest positive integer that is divisible by all given integers.


  • Lucas numbers Show Notations de

    The Lucas numbers are defined as follows:

    L0http://mathhub.info/smglom/numbers/lucasnumbers.omdoc?lucasnumbers?Lucas-numbers equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal 2
    L1http://mathhub.info/smglom/numbers/lucasnumbers.omdoc?lucasnumbers?Lucas-numbers equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal 1
    Lnhttp://mathhub.info/smglom/numbers/lucasnumbers.omdoc?lucasnumbers?Lucas-numbersn equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal (L(n-1))+(L(n-2))http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numbers/lucasnumbers.omdoc?lucasnumbers?Lucas-numbershttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numbers/lucasnumbers.omdoc?lucasnumbers?Lucas-numbershttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn

  • Lucas polynomials Show Notations de

    The Lucas polynomials are defined by the recurrence relation

    ((Ln(x))x)=(defined-piecewise(
    2wenn(n=0)
    )
    (
    xwenn(n=1)
    )
    (
    ((x(L(n-1)(x))x)+((L(n-2)(x))x))wenn(n2)
    )
    )
    http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/lucaspolynomials.omdoc?lucaspolynomials?Lucas-polynomialsnxhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecexhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationxhttp://mathhub.info/smglom/smglom/lucaspolynomials.omdoc?lucaspolynomials?Lucas-polynomialshttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/lucaspolynomials.omdoc?lucaspolynomials?Lucas-polynomialshttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnxhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methann

    The first few Lucas polynomials are:

    ((L0(x))x)=2http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/lucaspolynomials.omdoc?lucaspolynomials?Lucas-polynomialsx

    ((L1(x))x)=xhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/lucaspolynomials.omdoc?lucaspolynomials?Lucas-polynomialsxx

    ((L2(x))x)=((x2)+2)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/lucaspolynomials.omdoc?lucaspolynomials?Lucas-polynomialsxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationx

    ((L3(x))x)=((x3)+(3x))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/lucaspolynomials.omdoc?lucaspolynomials?Lucas-polynomialsxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationx

    ((L4(x))x)=((x4)+(4(x2))+2)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/lucaspolynomials.omdoc?lucaspolynomials?Lucas-polynomialsxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationx

    ((L5(x))x)=((x5)+(5(x3))+(5x))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/lucaspolynomials.omdoc?lucaspolynomials?Lucas-polynomialsxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationx


  • lucky numbers of Euler de

    The lucky numbers of Euler are positive integers nn such that ((x2)-x)+nhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationxxn is a prime number for x=0=1==(n-1)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn.


  • Lychrel number de

    A Lychrel number is a natural number that cannot form a palindromic number through the iterative process of repeatedly reversing its digits and adding the resulting numbers.


  • mantissa

    In scientific notation all numbers are written in the form of a×10bhttp://mathhub.info/smglom/numberfields/scinotation.omdoc?scinotation?scientific-notationab, (ahttp://mathhub.info/smglom/sets/set.omdoc?set?insetahttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number times 10 raised to the power of bhttp://mathhub.info/smglom/sets/set.omdoc?set?insetbhttp://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?integers), aa is called the significand or mantissa.

    a×10bhttp://mathhub.info/smglom/numberfields/scinotation.omdoc?scinotation?scientific-notationab is called normalized, iff 1(|a|)<10http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?betweeneshttp://mathhub.info/smglom/numberfields/absolutevalue.omdoc?absolutevalue?absolute-valuea


  • matrix Show Notations

    Let VV be a vector space over a field KK, then a matrix is a rectangular arrangement of members of KK.


  • maximum Show Notations

    The minimum (minimumS)http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?minimumS (maximum (maximumS)http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?maximumS) of an ordered set SS is that element mm (if it exists), such that all other membes of SS are smaller (larger) than mm.

    If ee is an expression and φφ a condition (in a variable xx), we write (bmaxvalφ[x]e)http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?bmaxvalφxe for (maximum(bsetst[x]e))http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?maximumhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstxe and call it the maximum for ee over φφ. Analogously, we write (bminvalφ[x]e)http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?bminvalφxe for (minimum(bsetst[x]φ))http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?minimumhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstxφ and call it the minimum for ee over φφ


  • maximum Show Notations

    The minimum (minimumS)http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?minimumS (maximum (maximumS)http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?maximumS) of an ordered set SS is that element mm (if it exists), such that all other membes of SS are smaller (larger) than mm.

    If ee is an expression and φφ a condition (in a variable xx), we write (bmaxvalφ[x]e)http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?bmaxvalφxe for (maximum(bsetst[x]e))http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?maximumhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstxe and call it the maximum for ee over φφ. Analogously, we write (bminvalφ[x]e)http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?bminvalφxe for (minimum(bsetst[x]φ))http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?minimumhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstxφ and call it the minimum for ee over φφ


  • maximum length de

    For a graph GG, the minimum length of a cycle contained in GG is the girth g(G)http://mathhub.info/smglom/graphs/graphcycleparameters.omdoc?graphcycleparameters?girthG of GG. The maximum length of a cycle contained in GG is the circumference of GG. For a graph which does not contain a cycle the girth is set to (g(G))=http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/graphs/graphcycleparameters.omdoc?graphcycleparameters?girthGhttp://mathhub.info/smglom/numberfields/infinity.omdoc?infinity?infinity, its circumference is set to zero.


  • Mclaurin series Show Notations de

    Let ff be a real-or complex-valued function that is smooth at a limit point aa of the domain of ff, then we call the infinite series given by

    (fundefeq[fxa]𝑇f(x;a))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfxahttp://mathhub.info/smglom/calculus/Taylor-series.omdoc?Taylor-series?Taylor-seriesfxa

    the Taylor series for ff around aa. If a=0http://mathhub.info/smglom/mv/equal.omdoc?equal?equala, then the series is known as the Mclaurin series.


  • Meertens number de

    A Meertens number is an integer that is its own Gödel number. Only one Meertens number is known:

    81312000=((28)(31)(53)(71)(112)(130)(170)(190))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiation

  • Mercator series de

    The Mercator series or Newton-Mercator series is the Taylor series for the natural logarithm:

    ln((1+x))http://mathhub.info/smglom/calculus/naturallogarithm.omdoc?naturallogarithm?natural-logarithmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionx

    The series converges whenever (multi-relation-expression1lethanxlessthan1)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanxhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthan.


  • Mersenne number Show Notations de

    A Mersenne number is a natural number of the form (Mn)=((2n)-1)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/mersennenumber.omdoc?mersennenumber?Mersenne-numbernhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn


  • Mersenne prime Show Notations de

    A Mersenne prime is a prime number of the form (Mn)=((2n)-1)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/primes/mersenneprime.omdoc?mersenneprime?Mersenne-primenhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn. Then nn is prime too.


  • Mertens function

    Mertens function is defined for all positive integers nn as MM where μkhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationμk is the Moebius function. The function is named in honour of Franz Mertens.

    The above definition can be extended to real numbers as follows:

    MM

  • metric de

    Let MM be a set, then we call a function d:(M×M)http://mathhub.info/smglom/sets/functions.omdoc?functions?fundhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsMMhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number a distance function (or metric ) on MM, iff for all x,y,zMhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetxyzM the following three identities hold:

    1. 1.

      (d)=0http://mathhub.info/smglom/mv/equal.omdoc?equal?equald iff x=yhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalxy (identity of indiscernibles),

    2. 2.

      (d)=(d)http://mathhub.info/smglom/mv/equal.omdoc?equal?equaldd (symmetry), and

    3. 3.

      (d)((d)+(d))http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethandhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additiondd (triangle inequality).

    We call M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd a metric space with base set MM and metric dd.


  • metric ) de

    Let MM be a set, then we call a function d:(M×M)http://mathhub.info/smglom/sets/functions.omdoc?functions?fundhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsMMhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number a distance function (or metric ) on MM, iff for all x,y,zMhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetxyzM the following three identities hold:

    1. 1.

      (d)=0http://mathhub.info/smglom/mv/equal.omdoc?equal?equald iff x=yhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalxy (identity of indiscernibles),

    2. 2.

      (d)=(d)http://mathhub.info/smglom/mv/equal.omdoc?equal?equaldd (symmetry), and

    3. 3.

      (d)((d)+(d))http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethandhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additiondd (triangle inequality).

    We call M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd a metric space with base set MM and metric dd.


  • metric space de

    Let MM be a set, then we call a function d:(M×M)http://mathhub.info/smglom/sets/functions.omdoc?functions?fundhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsMMhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number a distance function (or metric ) on MM, iff for all x,y,zMhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetxyzM the following three identities hold:

    1. 1.

      (d)=0http://mathhub.info/smglom/mv/equal.omdoc?equal?equald iff x=yhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalxy (identity of indiscernibles),

    2. 2.

      (d)=(d)http://mathhub.info/smglom/mv/equal.omdoc?equal?equaldd (symmetry), and

    3. 3.

      (d)((d)+(d))http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethandhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additiondd (triangle inequality).

    We call M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd a metric space with base set MM and metric dd.


  • minimum Show Notations

    The minimum (minimumS)http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?minimumS (maximum (maximumS)http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?maximumS) of an ordered set SS is that element mm (if it exists), such that all other membes of SS are smaller (larger) than mm.

    If ee is an expression and φφ a condition (in a variable xx), we write (bmaxvalφ[x]e)http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?bmaxvalφxe for (maximum(bsetst[x]e))http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?maximumhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstxe and call it the maximum for ee over φφ. Analogously, we write (bminvalφ[x]e)http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?bminvalφxe for (minimum(bsetst[x]φ))http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?minimumhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstxφ and call it the minimum for ee over φφ


  • minimum Show Notations de

    The minimum (minimumS)http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?minimumS (maximum (maximumS)http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?maximumS) of an ordered set SS is that element mm (if it exists), such that all other membes of SS are smaller (larger) than mm.

    If ee is an expression and φφ a condition (in a variable xx), we write (bmaxvalφ[x]e)http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?bmaxvalφxe for (maximum(bsetst[x]e))http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?maximumhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstxe and call it the maximum for ee over φφ. Analogously, we write (bminvalφ[x]e)http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?bminvalφxe for (minimum(bsetst[x]φ))http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?minimumhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstxφ and call it the minimum for ee over φφ


  • minimum length de

    For a graph GG, the minimum length of a cycle contained in GG is the girth g(G)http://mathhub.info/smglom/graphs/graphcycleparameters.omdoc?graphcycleparameters?girthG of GG. The maximum length of a cycle contained in GG is the circumference of GG. For a graph which does not contain a cycle the girth is set to (g(G))=http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/graphs/graphcycleparameters.omdoc?graphcycleparameters?girthGhttp://mathhub.info/smglom/numberfields/infinity.omdoc?infinity?infinity, its circumference is set to zero.


  • modulus

    The integer division operator computes the integer quotient (or modulus) ndivmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divnm of two natural numbers. ndivmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divnm is defined as that qhttp://mathhub.info/smglom/sets/set.omdoc?set?insetqhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, such that n=((mq)+r)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationmqr for some 0r<mhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?betweenesrm. The number rr is called the remainder and is written as nmodmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?modnm.


  • Motzkin number Show Notations de

    Motzkin numbers Mnhttp://mathhub.info/smglom/numbers/motzkinnumberrec.omdoc?motzkinnumberrec?Motzkin-numbern are defind by the recurrence relation:

    M(n+1)http://mathhub.info/smglom/numbers/motzkinnumberrec.omdoc?motzkinnumberrec?Motzkin-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionn

    where (multi-relation-expressionM0equalM1equal1)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numbers/motzkinnumberrec.omdoc?motzkinnumberrec?Motzkin-numberhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/motzkinnumberrec.omdoc?motzkinnumberrec?Motzkin-numberhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal.


  • Motzkin number

    Motzkin numbers Mnhttp://mathhub.info/smglom/numbers/motzkinnumbercat.omdoc?motzkinnumbercat?motzkinnumbercatn for natural numbers nn can be expressed in terms of Catalan numbers Ckhttp://mathhub.info/smglom/numbers/catalannumber.omdoc?catalannumber?catalannumberk:

    Mnhttp://mathhub.info/smglom/numbers/motzkinnumbercat.omdoc?motzkinnumbercat?motzkinnumbercatn

  • Motzkin number Show Notations de

    The Motzkin number Mnhttp://mathhub.info/smglom/numbers/motzkinnumber.omdoc?motzkinnumber?Motzkin-numbern for a given number nn is the number of different ways of drawing non-intersecting chords on a circle between nn points.

    The first Motzkin numbers are 1,1,2,4,9,21,51,127,323,835,2188,5798,...http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?infseq.


  • Motzkin prime de

    A Motzkin prime is a Motzkin number that is prime.


  • multi-relation expression Show Notations de

    A multi-relation expression is built up from binary relations via conjunction: (multi-relation-expressionaRbSc)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionaRbShttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationc holds, iff (R)R holds and also (multi-relation-expressionbSc)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionbSc.


  • multiperfect de

    A number nn is called multiperfect or kk-perfect for a given natural numbers kk, if and only if the sum of all positive divisors of nn is equal to knhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationkn.


  • multiple de

    Let Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR be a ring and m0http://mathhub.info/smglom/mv/equal.omdoc?equal?notequalm, then we say that mm is a left divisor of nn (or mm left divides nn) and that nn is a left multiple of mm, if there is a kRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetkR such that (multiplication)=nhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationn.

    Analogously we say that mm is a right divisor of nn (or mm right divides nn) and that nn is a right multiple of mm, if there is a kRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetkR such that (multiplication)=nhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationn.

    If RR is commxutative, then left and right divisors coincide and we simply speak of divisor and multiple and write m|nhttp://mathhub.info/smglom/algebra/ring-divisor.omdoc?ring-divisor?divisormn.

    1, 1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction, nn, and nhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn are known as the trivial divisors of nn. A divisor of nn that is not a trivial divisor is known as a proper or non-trivial divisor.


  • multiple Harshad number de

    A multiple Harshad number is a Harshad number that, when divided by the sum of its digits, produces another Harshad number.


  • multiplicative digital root de

    The multiplicative persistence is the number of steps required to reach a single digit.

    The multiplicative digital root of a positive integer nn is found by multiplying the digits of nn together, then repeating this operation until only a single digit remains. This single-digit number is called the multiplicative digital root of nn.


  • multiplicative persistence de

    The multiplicative persistence is the number of steps required to reach a single digit.


  • multiplicative structure de

    A structure Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR is called a ring, if Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR is a Abelian group (called the additive structure), Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR is a monoid (called the multiplicative structure), and Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR is a ringoid.

    We call 0 the zero of the ring and 1 the one of the ring.


  • mutually disjoint ro de tr

    A family of sets is called pairwise disjoint or mutually disjoint, if any two of them are disjoint.


  • mutually prime Show Notations de

    Two integers are said to be coprime (also spelled co-prime), relatively prime or mutually prime if their greatest common divisor is 1.


  • Münchhausen number

    A Münchhausen number is a number that is equal to the sum of its digits powered to itself.

    If n=(akak-1a0)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6ef is the decimal representation of nn then

    nn

  • n-minex

  • n-plex

    n-plex is defind as the integer 10nhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn.


  • Napier’s constant Show Notations de

    The number ee, sometimes called Euler’s constant (also known as Napier’s constant) is an important mathematical constant.

    ehttp://mathhub.info/smglom/numberfields/eulersnumber.omdoc?eulersnumber?eulersnumber
    e=2.718281828459045...http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?startswithhttp://mathhub.info/smglom/numberfields/eulersnumber.omdoc?eulersnumber?eulersnumber

  • Narayana number Show Notations de

    The Narayana numbers N(n,k)http://mathhub.info/smglom/numbers/narayananumber.omdoc?narayananumber?Narayana-numbernk count the number of paths from 0,0http://mathhub.info/smglom/sets/pair.omdoc?pair?pair to (2n),0http://mathhub.info/smglom/sets/pair.omdoc?pair?pairhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationn, with steps only northeast and southeast, not straying below the xx-axis, with kk peaks(maxima).

    (N(n,k))=((1n)(𝒞kn)(𝒞(k-1)n))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/narayananumber.omdoc?narayananumber?Narayana-numbernkhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionnhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficientnkhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficientnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionk

    where nn and kk are natural number with k1nhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?betweeneekn.


  • natural density de

    A subset AA of positive integers has an asymptotic density (or natural density) d(A)http://mathhub.info/smglom/smglom/asymptoticdensity.omdoc?asymptoticdensity?asymptotic-densityA, where (multi-relation-expression0lessthand(A)lessthan1)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthanhttp://mathhub.info/smglom/smglom/asymptoticdensity.omdoc?asymptoticdensity?asymptotic-densityAhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthan, if the limit exists

    d(A)http://mathhub.info/smglom/smglom/asymptoticdensity.omdoc?asymptoticdensity?asymptotic-densityA

    anhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationan is the number of elements of AA less than or equal to nn.


  • natural logarithm Show Notations de

    The natural logarithm is the logarithm to the base ehttp://mathhub.info/smglom/numberfields/eulersnumber.omdoc?eulersnumber?eulersnumber.


  • natural number Show Notations de

    The set http://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number of natural numbers is the set {(0,1,2,)}http://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?aseqdots.

    The set http://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?positive-natural-number of positive natural numbers is the set {(1,2,3,)}http://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?aseqdots.


  • near Wilson prime de

  • negative integers Show Notations de

    The set negative integers of http://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?negative-integers is the set {(,(-(3)),(-(2)),(-(1)))}http://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?dotsaseqhttp://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?uminushttp://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?uminushttp://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?uminus.


  • negative real number Show Notations de

    The set http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number of real numbers is defined as the completion of http://mathhub.info/smglom/numberfields/rationalnumbers.omdoc?rationalnumbers?rational-numbers.

    We use http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?negative-real-number and http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?positive-real-number for the sets of negative real numbers and positive real numbers.


  • neighborhood de

    Let X,Ohttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureXOMFOREIGNscala.xml.Node$@6040b6ef be a topological space and pXhttp://mathhub.info/smglom/sets/set.omdoc?set?insetpX, then we call any open set NOhttp://mathhub.info/smglom/sets/set.omdoc?set?insetNOMFOREIGNscala.xml.Node$@6040b6ef a neighborhood of pp, iff pNhttp://mathhub.info/smglom/sets/set.omdoc?set?insetpN.


  • neighbour de

    Two distinct edges ee and ff are adjacent if they have an end in common.

    Two vertices xx and yy in a graph GG are adjacent, or neighbours, if x,yhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairxy is an edge of GG.


  • Newman-Shanks-Williams prime de

    A Newman-Shanks-Williams prime, or NSW prime, is a prime number pp which can be written in the form

    p=((((1+(2))((2m)+1))+((1-(2))((2m)+1)))2)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?square-roothttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?square-roothttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationm

    where mm is a natural numbers.


  • Newton-Mercator series de

    The Mercator series or Newton-Mercator series is the Taylor series for the natural logarithm:

    ln((1+x))http://mathhub.info/smglom/calculus/naturallogarithm.omdoc?naturallogarithm?natural-logarithmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionx

    The series converges whenever (multi-relation-expression1lethanxlessthan1)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanxhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthan.


  • Newton’s de

    If f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN is differentiable on MM, then the derivative function (Dxf):MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfxMN of ff (with respect to xx) is defined as

    (fundefeq[fxa]Dxf(a))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfxahttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa

    The dependency of Dxfhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfx on xx is left implicit in this notation (Lagrange’s notation). In Leibniz’s notation we write the derivative of ff with respect to xx as Dxfhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfx and the derivative of ff at aa as Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa or Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa. In Euler’s notation, this is written as Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa and in Newton’snotation as Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa.


  • ninth smarandache constant Show Notations

    The ninth smarandache constant http://mathhub.info/smglom/smglom/smarandacheconstant1.omdoc?smarandacheconstant1?first-smarandache-constant is defind by

    http://mathhub.info/smglom/smglom/smarandacheconstant9.omdoc?smarandacheconstant9?ninth-smarandache-constant

    where S(n)http://mathhub.info/smglom/smglom/smarandachefunction.omdoc?smarandachefunction?smarandache-functionn is the smarandache function.

    The sum converges.


  • Niven number de

    A Harshad number or Niven number in a given number base is an integer that is divisible by the sum of its digits when written in that base.


  • Niven’s constant de

    We define (H1)=1http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationH and Hnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationHn is the largest exponent appearing in the prime factorization of a natural number n>1http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethann.

    Then Niven’s constant is given by

    limn(1n)http://mathhub.info/smglom/calculus/sequenceConvergence.omdoc?sequenceConvergence?limitnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionn

  • node de

    A directed graph (also called digraph or oriented graph) is a pair V,Ehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE such that VV is a set and E(V×V)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetEhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsVV. We call VV the vertices (or nodes) and EE the edges of GG.


  • node de

    A graph is a pair V,Ehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE such that VV is a set and E(V×V)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetEhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsVV is a subset of the set of pairs from VV. We call VV the vertices (or nodes, points,junctions) and EE the edges (or lines, branches, arcs) of GG.


  • non-trivial divisor de

    1, 1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction, nn, and nhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn are known as the trivial divisors of nn. A divisor of nn that is not a trivial divisor is known as a proper or non-trivial divisor.


  • normal

    xx is said to be normal in base bb if its digits in base bb follow a uniform distribution.

    A real number xx is said to be normal if its digits in every base follow a uniform distribution: all digits are equally likely, all pairs of digits equally likely, all triplets of digits equally likely, etc.


  • normal in base de

    xx is said to be normal in base bb if its digits in base bb follow a uniform distribution.


  • normalized

    In scientific notation all numbers are written in the form of a×10bhttp://mathhub.info/smglom/numberfields/scinotation.omdoc?scinotation?scientific-notationab, (ahttp://mathhub.info/smglom/sets/set.omdoc?set?insetahttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number times 10 raised to the power of bhttp://mathhub.info/smglom/sets/set.omdoc?set?insetbhttp://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?integers), aa is called the significand or mantissa.

    a×10bhttp://mathhub.info/smglom/numberfields/scinotation.omdoc?scinotation?scientific-notationab is called normalized, iff 1(|a|)<10http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?betweeneshttp://mathhub.info/smglom/numberfields/absolutevalue.omdoc?absolutevalue?absolute-valuea


  • NSW prime de

    A Newman-Shanks-Williams prime, or NSW prime, is a prime number pp which can be written in the form

    p=((((1+(2))((2m)+1))+((1-(2))((2m)+1)))2)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?square-roothttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?square-roothttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationm

    where mm is a natural numbers.


  • nullary product Show Notations de

    An empty product, or nullary product, is the result of multiplying no factors. It is equal to the multiplicative identity 1.


  • nullary sum Show Notations de

    An empty sum, or nullary sum, is a summation involving no terms at all. The value of any empty sum of numbers is conventionally taken to be zero.

    For example, if nmhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethannm then

    i=mnaihttp://mathhub.info/smglom/numberfields/sum.omdoc?sum?summniOMFOREIGNscala.xml.Node$@6040b6ef

  • number of edges Show Notations de

    The number of edges of a graph GG is written as Ghttp://mathhub.info/smglom/graphs/graphnumberofedges.omdoc?graphnumberofedges?graphnumberofedgesG.


  • number of vertices Show Notations de

    The number of vertices of a graph GG, called its order, is written as |G|http://mathhub.info/smglom/graphs/graphnumberofvertices.omdoc?graphnumberofvertices?graphnumberofverticesG.


  • Number theory de

    Number theory is a branch of pure mathematics devoted primarily to the study of the integers.


  • oblong number Show Notations de

    A pronic number Pnhttp://mathhub.info/smglom/numbers/pronicnumber.omdoc?pronicnumber?pronic-numbern, oblong number, rectangular number or heteromecic number, is a number which is the product of two consecutive natural numbers.

    (multi-relation-expressionPnequal(nn+1)equal(n2)+n)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numbers/pronicnumber.omdoc?pronicnumber?pronic-numbernhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationnn

  • on

    We call a triple P,L,Ihttp://mathhub.info/smglom/mv/structure.omdoc?structure?structurePLI an incidence structure (or incidence geometry), with points PP, lines LL, and incidence realation I(P×L)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetIhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsPL (we say pPhttp://mathhub.info/smglom/sets/set.omdoc?set?insetpP is on ll, if (p,l)Ihttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairplI), iff

    • There are at least two points in PP.

    • For any two points p,qPhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetpqP, there is exactly one line lLhttp://mathhub.info/smglom/sets/set.omdoc?set?insetlL such that (p,l)Ihttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairplI and (q,l)Ihttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairqlI.

    • For every line lLhttp://mathhub.info/smglom/sets/set.omdoc?set?insetlL there are at least two points on ll.

    • For every line lLhttp://mathhub.info/smglom/sets/set.omdoc?set?insetlL there is at least one point pp with plhttp://mathhub.info/smglom/sets/set.omdoc?set?ninsetpl.


  • one de

    A structure Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR is called a ring, if Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR is a Abelian group (called the additive structure), Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR is a monoid (called the multiplicative structure), and Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR is a ringoid.

    We call 0 the zero of the ring and 1 the one of the ring.


  • open ball Show Notations de

    Let M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd be a metric space, then we call the set (fundefeq[rx]𝐵(r,x))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqrxhttp://mathhub.info/smglom/calculus/ball.omdoc?ball?open-ballrx the open ball and (fundefeq[rx]B¯(r,x))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqrxhttp://mathhub.info/smglom/calculus/ball.omdoc?ball?closed-ballrx the closed ball around xx with radius rr. We also write 𝐵(r,x)http://mathhub.info/smglom/calculus/ball.omdoc?ball?open-ballrx and B¯(r,x)http://mathhub.info/smglom/calculus/ball.omdoc?ball?closed-ballrx.


  • open set de

    A topological space X,Ohttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureXOMFOREIGNscala.xml.Node$@6040b6ef is a set XX together with a collection O(𝒫(X))http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/sets/powerset.omdoc?powerset?powersetX, such that

    1. 1.

      ,XOhttp://mathhub.info/smglom/sets/set.omdoc?set?minsethttp://mathhub.info/smglom/sets/emptyset.omdoc?emptyset?esetXOMFOREIGNscala.xml.Node$@6040b6ef.

    2. 2.

      (SSs)Ohttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/union.omdoc?union?munionCollectionOMFOREIGNscala.xml.Node$@6040b6efSsOMFOREIGNscala.xml.Node$@6040b6ef if SShttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetSOMFOREIGNscala.xml.Node$@6040b6ef.

    3. 3.

      (SSs)Ohttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/intersection.omdoc?intersection?mintersectCollectionOMFOREIGNscala.xml.Node$@6040b6efSsOMFOREIGNscala.xml.Node$@6040b6ef if SShttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetSOMFOREIGNscala.xml.Node$@6040b6ef is finite.

    OOMFOREIGNscala.xml.Node$@6040b6ef is called an open set topology (or just topology) on XX. Members of a topology OOMFOREIGNscala.xml.Node$@6040b6ef are called open sets and their complements closed sets. A subset of XX may be neither closed nor open, either closed or open, or both. A set that is both closed and open is called a clopen set.


  • open set topology de

    A topological space X,Ohttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureXOMFOREIGNscala.xml.Node$@6040b6ef is a set XX together with a collection O(𝒫(X))http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/sets/powerset.omdoc?powerset?powersetX, such that

    1. 1.

      ,XOhttp://mathhub.info/smglom/sets/set.omdoc?set?minsethttp://mathhub.info/smglom/sets/emptyset.omdoc?emptyset?esetXOMFOREIGNscala.xml.Node$@6040b6ef.

    2. 2.

      (SSs)Ohttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/union.omdoc?union?munionCollectionOMFOREIGNscala.xml.Node$@6040b6efSsOMFOREIGNscala.xml.Node$@6040b6ef if SShttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetSOMFOREIGNscala.xml.Node$@6040b6ef.

    3. 3.

      (SSs)Ohttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/intersection.omdoc?intersection?mintersectCollectionOMFOREIGNscala.xml.Node$@6040b6efSsOMFOREIGNscala.xml.Node$@6040b6ef if SShttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetSOMFOREIGNscala.xml.Node$@6040b6ef is finite.

    OOMFOREIGNscala.xml.Node$@6040b6ef is called an open set topology (or just topology) on XX. Members of a topology OOMFOREIGNscala.xml.Node$@6040b6ef are called open sets and their complements closed sets. A subset of XX may be neither closed nor open, either closed or open, or both. A set that is both closed and open is called a clopen set.


  • order de

    The number of vertices of a graph GG, called its order, is written as |G|http://mathhub.info/smglom/graphs/graphnumberofvertices.omdoc?graphnumberofvertices?graphnumberofverticesG.


  • order geometry

    We call a set of points a ternary relation betweenhttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?between an order geometry , iff

    1. 1.

      If A*B*Chttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?betweenABC, then

      1. (a)

        AChttp://mathhub.info/smglom/mv/equal.omdoc?equal?notequalAC.

      2. (b)

        {A,B,C}http://mathhub.info/smglom/sets/set.omdoc?set?setABC is a collinear set.

      3. (c)

        B*B*Ahttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?betweenBBA.

    2. 2.

      If BB and DD are distinct points, then there is at least one point AA, such that A*B*Dhttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?betweenABD.

    3. 3.

      If AA, BB and CC are distinct collinear points, then A*B*Chttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?betweenABC, or A*C*Bhttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?betweenACB, or B*A*Chttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?betweenBAC.

    4. 4.

      A*B*Chttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?betweenABC and A*C*Bhttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?betweenACB cannot hold at the same time.

    If A*B*Chttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?betweenABC, we say that BB is between AA and CC.


  • ordered set de

    We call a partial ordering RR a total ordering (or simple order or linear order), iff abhttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?poleab or bahttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?poleba for all a,bAhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabA.

    We call a structure S,polehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureShttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole of a set SS and a total ordering polehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole an totally ordered set.

    We call a structure S,polehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureShttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole of a set SS and a partial ordering polehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole an ordered set.


  • Ore number de

    A harmonic divisor number, or Ore number, is a positive integer whose divisors have a harmonic mean that is an integer.


  • oriented graph de

    A directed graph (also called digraph or oriented graph) is a pair V,Ehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE such that VV is a set and E(V×V)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetEhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsVV. We call VV the vertices (or nodes) and EE the edges of GG.


  • outdegree

    Let G=(V,E)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalGhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE be a directed graph and vVhttp://mathhub.info/smglom/sets/set.omdoc?set?insetvV a vertex in GG, then we define

    • indegree indeg(v)http://mathhub.info/smglom/graphs/inout-degree.omdoc?inout-degree?indegv of vv as #((bsetst[w]w))http://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstww

    • outdegree outdeg(v)http://mathhub.info/smglom/graphs/inout-degree.omdoc?inout-degree?outdegv of vv as #((bsetst[w]w))http://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstww


  • outer limit Show Notations

    Ist M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd ein metrischer Raum und (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan eine Folge mit (ai)Mhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselaiM für ihttp://mathhub.info/smglom/sets/set.omdoc?set?insetihttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, so ist der

    • Limes superior limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (schreibe auch limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) definiert durch

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai
    • Limes inferior limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (schreibe auch limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) definiert durch

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai

    Let M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd be a metric space and (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan a sequence with (ai)Mhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselaiM for all ihttp://mathhub.info/smglom/sets/set.omdoc?set?insetihttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, then the

    • limit superior (also called supremum limit, superior limit, upper limit, or outer limit) limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (also written as limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) is defined as

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai
    • limit inferior (also called infimum limit, inferior limit, lower limit, or inner limit) limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (also written as limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) is defined as

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai

  • Padovan number de

    The Padovan numbers are defined by the recurrence relation

    (multi-relation-expressionP0equalP1equalP2equal1)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numbers/padovannumbers.omdoc?padovannumbers?padovannumbershttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/padovannumbers.omdoc?padovannumbers?padovannumbershttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/padovannumbers.omdoc?padovannumbers?padovannumbershttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal
    (Pn)=((P(n-2))+(P(n-3)))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/padovannumbers.omdoc?padovannumbers?padovannumbersnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numbers/padovannumbers.omdoc?padovannumbers?padovannumbershttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numbers/padovannumbers.omdoc?padovannumbers?padovannumbershttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn

  • pair Show Notations tr ro de

    Let AA and BB be sets, then the set of pairs A×Bhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAB of AA and BB is defined as (bsetst[ab]a,b)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstabhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairab, we call (a,b)(A×B)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairabhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAB a pair.


  • pairs Show Notations tr ro de

    Let AA and BB be sets, then the set of pairs A×Bhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAB of AA and BB is defined as (bsetst[ab]a,b)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstabhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairab, we call (a,b)(A×B)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairabhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAB a pair.


  • pairwise disjoint ro de tr

    A family of sets is called pairwise disjoint or mutually disjoint, if any two of them are disjoint.


  • panarithmic number de

    A practical number or panarithmic number is a positive integer nn such that all smaller positive integers can be represented as sums of distinct divisors of nn.


  • parent

    A tree is a directed acyclic graph G=(V,E)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalGhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE such that

    • there is exactly one initialnode vrVhttp://mathhub.info/smglom/sets/set.omdoc?set?insetOMFOREIGNscala.xml.Node$@6040b6efV (called the root), and

    • all nodes but the root have indegree 1.

    We call vv the parent of ww, iff (v,w)Ehttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairvwE (ww is a child of vv). We call a node vv a leaf of GG, iff it is terminal, i.e. if it does not have children.


  • partial function de

    A relation f(X×Y)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetfhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsXY, is called a partial function with domain XX (write 𝐝𝐨𝐦(f)http://mathhub.info/smglom/sets/functions.omdoc?functions?domainf) and codomain YY (write 𝐜𝐨𝐝𝐨𝐦(f)http://mathhub.info/smglom/sets/functions.omdoc?functions?codomainf), iff for all xXhttp://mathhub.info/smglom/sets/set.omdoc?set?insetxX there is at most one yYhttp://mathhub.info/smglom/sets/set.omdoc?set?insetyY with (x,y)fhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairxyf.

    We write f:XY;xyhttp://mathhub.info/smglom/sets/functions.omdoc?functions?partfunsuchthatfXYxy and (f)=yhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalfy instead of (x,y)fhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairxyf.


  • partial order de

    We call a partial ordering RR a total ordering (or simple order or linear order), iff abhttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?poleab or bahttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?poleba for all a,bAhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabA.

    We call a structure S,polehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureShttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole of a set SS and a total ordering polehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole an totally ordered set.

    We call a preorder pole(A×A)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsethttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?polehttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAA on AA a partial ordering (or partial order), iff it is antisymmetric. We associate with polehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole a strict ordering poless:=(bsetst[ab](a,b)pole)http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?polesshttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstabhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairabhttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole.

    We often also use the converse relations pomehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pome and pomorehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pomore.

    We call a structure S,polehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureShttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole of a set SS and a partial ordering polehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole an ordered set.


  • partial ordering de

    We call a partial ordering RR a total ordering (or simple order or linear order), iff abhttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?poleab or bahttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?poleba for all a,bAhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabA.

    We call a structure S,polehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureShttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole of a set SS and a total ordering polehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole an totally ordered set.

    We call a preorder pole(A×A)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsethttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?polehttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAA on AA a partial ordering (or partial order), iff it is antisymmetric. We associate with polehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole a strict ordering poless:=(bsetst[ab](a,b)pole)http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?polesshttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstabhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairabhttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole.

    We often also use the converse relations pomehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pome and pomorehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pomore.

    We call a structure S,polehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureShttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole of a set SS and a partial ordering polehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole an ordered set.


  • partial sum Show Notations de

    For any sequence (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan with (an)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselanhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number or (an)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselanhttp://mathhub.info/smglom/numberfields/complexnumbers.omdoc?complexnumbers?complex-number we define the nn-th partial sum (fundefeq[an]sn)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqanhttp://mathhub.info/smglom/calculus/series.omdoc?series?partial-suman.

    The series induced by (na)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequencean is the sequence of partial sums (sequenceon[an]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonann.


  • path Show Notations de

    Given a directed graph G=(V,E)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalGhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE we call a vector (p=((v0,,vn)))(V(n+1))http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalphttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlivnhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?nCartSpaceVhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionn a path in GG iff (vi-1,vi)Ehttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6efE for all 1inhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?betweeneein, n>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethann.

    • v0OMFOREIGNscala.xml.Node$@6040b6ef is called the start of pp (write start(p)http://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pstartp )

    • vnOMFOREIGNscala.xml.Node$@6040b6ef is called the end of pp (write end(p)http://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pendp )

    • nn is called the length of pp (write len(p)http://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?plenp )


  • Pell number de

    The Pell numbers are defined by the recurrence relation

    P0http://mathhub.info/smglom/numbers/pellnumbers.omdoc?pellnumbers?pellnumbers equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal 0
    P1http://mathhub.info/smglom/numbers/pellnumbers.omdoc?pellnumbers?pellnumbers equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal 1
    Pnhttp://mathhub.info/smglom/numbers/pellnumbers.omdoc?pellnumbers?pellnumbersn equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal (2(P(n-1)))+(P(n-2))http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numbers/pellnumbers.omdoc?pellnumbers?pellnumbershttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numbers/pellnumbers.omdoc?pellnumbers?pellnumbershttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn

  • perfect number de

    A perfect number is a positive integer that is equal to the sum of its aliquot sum.


  • permutable prime de

    A permutable prime is a prime that is prime for all permutations of its digits.

    For example:

    131

  • Perrin number de

    The Perrin numbers are defined by the recurrence relation

    P0http://mathhub.info/smglom/numbers/perrinnumbers.omdoc?perrinnumbers?perrinnumbers equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal 3
    P1http://mathhub.info/smglom/numbers/perrinnumbers.omdoc?perrinnumbers?perrinnumbers equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal 0
    P2http://mathhub.info/smglom/numbers/perrinnumbers.omdoc?perrinnumbers?perrinnumbers equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal 2
    Pnhttp://mathhub.info/smglom/numbers/perrinnumbers.omdoc?perrinnumbers?perrinnumbersn equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal (P(n-2))+(P(n-3))http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numbers/perrinnumbers.omdoc?perrinnumbers?perrinnumbershttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numbers/perrinnumbers.omdoc?perrinnumbers?perrinnumbershttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn

  • Perrin pseudoprime de

    A Perrin pseudoprime is a composite number nn that divides the Perrin number Pnhttp://mathhub.info/smglom/numbers/perrinpseudoprime.omdoc?perrinpseudoprime?perrinpseudoprimen.


  • Pierpont prime de

    A Pierpont prime is a prime number of the form

    ((2u)(3v))+1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationuhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationv

    for some nonnegative integers uu and vv.


  • plastic constant Show Notations de

    The plastic number ρhttp://mathhub.info/smglom/smglom/plasticnumber.omdoc?plasticnumber?plastic-number (also known as the plastic constant) is a mathematical constant which is the unique real solution of the cubic equation (x3)=(x+1)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionx. It has the value

    ρ=((((108+(12(69)))3)+((108-(12(69)))3))6)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/smglom/plasticnumber.omdoc?plasticnumber?plastic-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?roothttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?square-roothttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?roothttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?square-root
    ρ=1.324717957244746...http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?startswithhttp://mathhub.info/smglom/smglom/plasticnumber.omdoc?plasticnumber?plastic-number

  • plastic number Show Notations de

    The plastic number ρhttp://mathhub.info/smglom/smglom/plasticnumber.omdoc?plasticnumber?plastic-number (also known as the plastic constant) is a mathematical constant which is the unique real solution of the cubic equation (x3)=(x+1)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionx. It has the value

    ρ=((((108+(12(69)))3)+((108-(12(69)))3))6)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/smglom/plasticnumber.omdoc?plasticnumber?plastic-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?roothttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?square-roothttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?roothttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?square-root
    ρ=1.324717957244746...http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?startswithhttp://mathhub.info/smglom/smglom/plasticnumber.omdoc?plasticnumber?plastic-number

  • point de

    A graph is a pair V,Ehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE such that VV is a set and E(V×V)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetEhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsVV is a subset of the set of pairs from VV. We call VV the vertices (or nodes, points,junctions) and EE the edges (or lines, branches, arcs) of GG.


  • point

    We call a triple P,L,Ihttp://mathhub.info/smglom/mv/structure.omdoc?structure?structurePLI an incidence structure (or incidence geometry), with points PP, lines LL, and incidence realation I(P×L)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetIhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsPL (we say pPhttp://mathhub.info/smglom/sets/set.omdoc?set?insetpP is on ll, if (p,l)Ihttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairplI), iff

    • There are at least two points in PP.

    • For any two points p,qPhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetpqP, there is exactly one line lLhttp://mathhub.info/smglom/sets/set.omdoc?set?insetlL such that (p,l)Ihttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairplI and (q,l)Ihttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairqlI.

    • For every line lLhttp://mathhub.info/smglom/sets/set.omdoc?set?insetlL there are at least two points on ll.

    • For every line lLhttp://mathhub.info/smglom/sets/set.omdoc?set?insetlL there is at least one point pp with plhttp://mathhub.info/smglom/sets/set.omdoc?set?ninsetpl.


  • pointwise convergent

    Let AA be a set, B,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureBd a metric space, and (sequenceon[f]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonfn a sequence of functions (fi):ABhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funhttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselfiAB, then we call (sequenceon[f]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonfn pointwise convergent to f:ABhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfAB on AAhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efA, iff limi(fi)http://mathhub.info/smglom/calculus/sequenceConvergence.omdoc?sequenceConvergence?limitihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselfi for all xAhttp://mathhub.info/smglom/sets/set.omdoc?set?insetxOMFOREIGNscala.xml.Node$@6040b6ef.

    Ist AA eine Menge, B,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureBd ein metrischer Raum und (sequenceon[f]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonfn eine Folge von Funktionen (fi):ABhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funhttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselfiAB, dann nennen wir (sequenceon[f]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonfn punktweise konvergent gegen f:ABhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfAB auf AAhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efA, falls limi(fi)http://mathhub.info/smglom/calculus/sequenceConvergence.omdoc?sequenceConvergence?limitihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselfi für alle xAhttp://mathhub.info/smglom/sets/set.omdoc?set?insetxOMFOREIGNscala.xml.Node$@6040b6ef.


  • polite number de

    A polite number is a positive integer that can be written as the sum of two or more consecutive positive integers.


  • positive natural number Show Notations de

    The set http://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?positive-natural-number of positive natural numbers is the set {(1,2,3,)}http://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?aseqdots.


  • positive real number Show Notations de

    The set http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number of real numbers is defined as the completion of http://mathhub.info/smglom/numberfields/rationalnumbers.omdoc?rationalnumbers?rational-numbers.

    We use http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?negative-real-number and http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?positive-real-number for the sets of negative real numbers and positive real numbers.


  • Poulet number de

    Poulet numbers are Fermat pseudoprimes to base 2.


  • powerful number de

    A powerful number is a positive integer mm such that for every prime number pp dividing mm, p2http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationp also divides mm.

    A powerful number is the product of a square and a cube. Powerful numbers are also known as squareful numbers, square-full numbers, or 2-full numbers.


  • pq number de

  • practical number de

    A practical number or panarithmic number is a positive integer nn such that all smaller positive integers can be represented as sums of distinct divisors of nn.


  • pre-image Show Notations de

    Let f:ABhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfAB be a function, AAhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efA, and BBhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efB, then we call

    • (fundefeq[fname.cvar.2]f(A))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfname.cvar.2http://mathhub.info/smglom/sets/image.omdoc?image?imageoffOMFOREIGNscala.xml.Node$@6040b6ef the image of AOMFOREIGNscala.xml.Node$@6040b6ef under ff,

    • (fundefeq[f]𝐈𝐦(f))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfhttp://mathhub.info/smglom/sets/image.omdoc?image?imagef the image of ff, and

    • (fundefeq[fname.cvar.6]f-1(B))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfname.cvar.6http://mathhub.info/smglom/sets/image.omdoc?image?pre-imagefOMFOREIGNscala.xml.Node$@6040b6ef the pre-image of BOMFOREIGNscala.xml.Node$@6040b6ef under ff.


  • preorder

    We call a structure S,rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureSr of a set SS equipped with a preorder rr an preordered set or proset.

    We call a binary relation r(A×A)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetrhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAA on AA a preorder (or quasiorder), iff it is reflexive and transitive.


  • preordered set

    We call a structure S,rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureSr of a set SS equipped with a preorder rr an preordered set or proset.


  • prime factor de

    A prime number pp is called a prime factor of the natural number nn, if pp is a divisor of nn.


  • prime gap Show Notations de

    A prime gap is the difference between two successive prime numbers. The nn-th prime gap, denoted gnhttp://mathhub.info/smglom/primes/primegap.omdoc?primegap?prime-gapn or g(pn)http://mathhub.info/smglom/primes/primegap.omdoc?primegap?prime-gaphttp://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?prime-numbern is the difference between the n+1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionn-th and the nn-th prime number.

    (gn)=((p(n+1))-(pn))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/primes/primegap.omdoc?primegap?prime-gapnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?prime-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnhttp://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?prime-numbern

  • prime number Show Notations de

    The number of prime numbers not greater than nn is written as π(n)http://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?NumberPrimeNumbern.

    A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.


  • prime quadruple de

    A prime quadruplet (sometimes called prime quadruple) is a set of four primes of the form {p,(p+2),(p+6),(p+8)}http://mathhub.info/smglom/sets/set.omdoc?set?setphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionp.

    For example:

    {5,7,11,13}http://mathhub.info/smglom/sets/set.omdoc?set?set
    {11,13,17,19}http://mathhub.info/smglom/sets/set.omdoc?set?set
    {101,103,107,109}http://mathhub.info/smglom/sets/set.omdoc?set?set

  • prime quadruplet de

    A prime quadruplet (sometimes called prime quadruple) is a set of four primes of the form {p,(p+2),(p+6),(p+8)}http://mathhub.info/smglom/sets/set.omdoc?set?setphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionp.

    For example:

    {5,7,11,13}http://mathhub.info/smglom/sets/set.omdoc?set?set
    {11,13,17,19}http://mathhub.info/smglom/sets/set.omdoc?set?set
    {101,103,107,109}http://mathhub.info/smglom/sets/set.omdoc?set?set

  • prime quintuplet de

    If {p,(p+2),(p+6),(p+8)}http://mathhub.info/smglom/sets/set.omdoc?set?setphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionp is a prime quadruplet and p-4http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionp or p+12http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionp is also prime, then the five primes form a prime quintuplet.

    For example:

    {5,7,11,13,17}http://mathhub.info/smglom/sets/set.omdoc?set?set
    {7,11,13,17,19}http://mathhub.info/smglom/sets/set.omdoc?set?set
    {11,13,17,19,23}http://mathhub.info/smglom/sets/set.omdoc?set?set

  • prime sextuplet de

    If {p,(p+2),(p+6),(p+8)}http://mathhub.info/smglom/sets/set.omdoc?set?setphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionp is a prime quadruplet and both p-4http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionp and p+12http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionp are also primes, then the six primes form a prime sextuplet.

    For example:

    {7,11,13,17,19,23}http://mathhub.info/smglom/sets/set.omdoc?set?set
    {97,101,103,107,109,113}http://mathhub.info/smglom/sets/set.omdoc?set?set
    {16057,16061,16063,16067,16069,16073}http://mathhub.info/smglom/sets/set.omdoc?set?set

  • prime twin de

    A prime twin is a pair of prime numbers with a difference of 2.


  • primitive Pythagorean triple de

    A Pythagorean triple consists of three natural numbers aa, bb and cc, such that ((a2)+(b2))=(c2)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationahttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationbhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationc. A primitive Pythagorean triple is one in which aa, bb and cc are coprime.


  • primitive root de

    A number gg is a primitive root modulo nn if every number coprime to nn is congruent modulo to a power of gg modulo nn.


  • primorial Show Notations de

    The primorial of a natural numbers nn, denoted by n#http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationn%23, is the product of all prime numbers less than or equal to nn.


  • product de

    We define the product over a sequence aiOMFOREIGNscala.xml.Node$@6040b6ef by

    (i=nmai):=(defined-piecewise(
    1wenn(n=m)
    )
    (
    (an)sonst
    )
    )
    http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdfromtonmiOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnmhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationOMFOREIGNscala.xml.Node$@6040b6ef

    The variable ii is called the index of multiplication and nn and mm the lower and upper bounds of the product respectively, together the specify the range of multiplication.

    There are variant product operators φaihttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdCollφOMFOREIGNscala.xml.Node$@6040b6ef and iSaihttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdInCollSiOMFOREIGNscala.xml.Node$@6040b6ef. The first one specifies the range of the product via a formula φφ in ii and the second one directly by giving a set SS.


  • projection Show Notations de

    Let AiOMFOREIGNscala.xml.Node$@6040b6ef be a collection of sets for 1inhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?betweeneein, then the nn-fold Cartesian product A1×...×Anhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?ncartliAn is (bsetst[an](a1,,an))http://mathhub.info/smglom/sets/set.omdoc?set?bsetstanhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlian, we call ((a1,,an))(A1×...×An)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlianhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?ncartliAn an nn-tuple.

    We call the function (projectioni):(A1×...×An)Ai;((a1,,an))aihttp://mathhub.info/smglom/sets/functions.omdoc?functions?funsuchthathttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?projectionihttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?ncartliAnOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlianOMFOREIGNscala.xml.Node$@6040b6ef the (ii th ) projection.


  • projection Show Notations de

    Let SS be a set and RR be an equivalence relation on SS, then for any we call xShttp://mathhub.info/smglom/sets/set.omdoc?set?insetxS we call the set (fundefeq[xR][x]R)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqxRhttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?equivalence-classxR the equivalence class of xx (under RR), and the set (fundefeq[xR]S_R)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqxRhttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?quotient-spaceSR the quotient space of SS (under RR).

    The mapping (projectionR):S(S_R);x([x]R)http://mathhub.info/smglom/sets/functions.omdoc?functions?funsuchthathttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?projectionRShttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?quotient-spaceSRxhttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?equivalence-classxR is called the projection of SS to S_Rhttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?quotient-spaceSR.


  • pronic number Show Notations de

    A pronic number Pnhttp://mathhub.info/smglom/numbers/pronicnumber.omdoc?pronicnumber?pronic-numbern, oblong number, rectangular number or heteromecic number, is a number which is the product of two consecutive natural numbers.

    (multi-relation-expressionPnequal(nn+1)equal(n2)+n)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numbers/pronicnumber.omdoc?pronicnumber?pronic-numbernhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationnn

  • proper de

    1, 1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction, nn, and nhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn are known as the trivial divisors of nn. A divisor of nn that is not a trivial divisor is known as a proper or non-trivial divisor.


  • proper subset Show Notations ro de tr

    A set AA is a proper subset of a set BB (written ABhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetAB), iff ABhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?proper-subsetAB but ABhttp://mathhub.info/smglom/sets/set.omdoc?set?nsetequalAB.


  • proper superset Show Notations ro tr

    A set AA is a proper superset of a set BB (written ABhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?supersetAB), iff BAhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetBA.


  • proset

    We call a structure S,rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureSr of a set SS equipped with a preorder rr an preordered set or proset.


  • Proth number de

    A Proth number is a number of the form (k(2n))+1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationkhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn where kk is an odd positive integer and nn is a positive integer such that (2n)>khttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationnk.

    The Cullen numbers (n2n)+1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn and Fermat numbers (2(2n))+1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn are special cases of Proth numbers.


  • Proth prime de

    A Proth prime is a prime number of the form (k2n)+1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionkhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn where kk is an odd positive integer and nn is a positive integer such that (2n)>khttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationnk.


  • punktweise konvergent

    Let AA be a set, B,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureBd a metric space, and (sequenceon[f]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonfn a sequence of functions (fi):ABhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funhttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselfiAB, then we call (sequenceon[f]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonfn pointwise convergent to f:ABhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfAB on AAhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efA, iff limi(fi)http://mathhub.info/smglom/calculus/sequenceConvergence.omdoc?sequenceConvergence?limitihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselfi for all xAhttp://mathhub.info/smglom/sets/set.omdoc?set?insetxOMFOREIGNscala.xml.Node$@6040b6ef.

    Ist AA eine Menge, B,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureBd ein metrischer Raum und (sequenceon[f]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonfn eine Folge von Funktionen (fi):ABhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funhttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselfiAB, dann nennen wir (sequenceon[f]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonfn punktweise konvergent gegen f:ABhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfAB auf AAhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efA, falls limi(fi)http://mathhub.info/smglom/calculus/sequenceConvergence.omdoc?sequenceConvergence?limitihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselfi für alle xAhttp://mathhub.info/smglom/sets/set.omdoc?set?insetxOMFOREIGNscala.xml.Node$@6040b6ef.


  • Pythagorean box de

    A Pythagorean box is a quadruple a,b,c,dhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tupleabcd of natural numbers those define a cuboid with side lengths aa, bb, cc and the space diagonal length dd.


  • Pythagorean prime de

    A Pythagorean prime is a prime number of the form

    (4n)+1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationn

  • Pythagorean quadruple de

    A Pythagorean quadruple (a,b,c,d)http://mathhub.info/smglom/smglom/pythagoreanquadruple.omdoc?pythagoreanquadruple?pythquadabcd is a tuple of integers aa, bb, cc and dd, such that d>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethand and

    ((a2)+(b2)+(c2))=(d2)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationahttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationbhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationchttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationd

  • Pythagorean triple de

    A Pythagorean triple consists of three natural numbers aa, bb and cc, such that ((a2)+(b2))=(c2)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationahttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationbhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationc. A primitive Pythagorean triple is one in which aa, bb and cc are coprime.


  • quasiorder

    We call a structure S,rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureSr of a set SS equipped with a preorder rr an preordered set or proset.

    We call a binary relation r(A×A)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetrhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAA on AA a preorder (or quasiorder), iff it is reflexive and transitive.


  • quotient space Show Notations de

    Let SS be a set and RR be an equivalence relation on SS, then for any we call xShttp://mathhub.info/smglom/sets/set.omdoc?set?insetxS we call the set (fundefeq[xR][x]R)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqxRhttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?equivalence-classxR the equivalence class of xx (under RR), and the set (fundefeq[xR]S_R)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqxRhttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?quotient-spaceSR the quotient space of SS (under RR).

    The mapping (projectionR):S(S_R);x([x]R)http://mathhub.info/smglom/sets/functions.omdoc?functions?funsuchthathttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?projectionRShttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?quotient-spaceSRxhttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?equivalence-classxR is called the projection of SS to S_Rhttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?quotient-spaceSR.


  • Ramanujan 6-10-8 identity de

    Let (ad)=(bc)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationadhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationbc for real numbers aa, bb, cc, and dd, then the Ramanujan 6-10-8 identity is given by

    ((64((((((a+b+c)6)+((b+c+d)6))-((c+d+a)6)-((d+a+b)6))+((a-d)6))-((b-c)6)))((((((a+b+c)10)+((b+c+d)10))-((c+d+a)10)-((d+a+b)10))+((a-d)10))-((b-c)10)))=(45(((((((a+b+c)8)+((b+c+d)8))-((c+d+a)8)-((d+a+b)8))+((a-d)8))-((b-c)8))2))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionabchttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionbcdhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additioncdahttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additiondabhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionadhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionbchttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionabchttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionbcdhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additioncdahttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additiondabhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionadhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionbchttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionabchttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionbcdhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additioncdahttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additiondabhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionadhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionbc

  • Ramanujan-Soldner constant Show Notations de

    The Ramanujan-Soldner constant is defined as the unique positive zero of the logarithmic integral li(x)http://mathhub.info/smglom/smglom/logarithmicintegralsmall.omdoc?logarithmicintegralsmall?logarithmicintsmallx.

    μ=1.451369234883381...http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?startswithhttp://mathhub.info/smglom/smglom/ramanujanSoldnerConstant.omdoc?ramanujanSoldnerConstant?Ramanujan-Soldner-constant

  • Ramanujan’s constant Show Notations de

    Ramanujan’s constant is the irrational number R:=(e(π(163)))http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/smglom/ramanujanconstant.omdoc?ramanujanconstant?Ramanujans-constanthttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/eulersnumber.omdoc?eulersnumber?eulersnumberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/pinumber.omdoc?pinumber?pinumberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?square-root. It is very close to an integer: R=2.62537412640768736E17...http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?startswithhttp://mathhub.info/smglom/smglom/ramanujanconstant.omdoc?ramanujanconstant?Ramanujans-constant.


  • range of multiplication de

    We define the product over a sequence aiOMFOREIGNscala.xml.Node$@6040b6ef by

    (i=nmai):=(defined-piecewise(
    1wenn(n=m)
    )
    (
    (an)sonst
    )
    )
    http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdfromtonmiOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnmhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationOMFOREIGNscala.xml.Node$@6040b6ef

    The variable ii is called the index of multiplication and nn and mm the lower and upper bounds of the product respectively, together the specify the range of multiplication.

    There are variant product operators φaihttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdCollφOMFOREIGNscala.xml.Node$@6040b6ef and iSaihttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdInCollSiOMFOREIGNscala.xml.Node$@6040b6ef. The first one specifies the range of the product via a formula φφ in ii and the second one directly by giving a set SS.


  • range of multiplication de

    We define the product over a sequence aiOMFOREIGNscala.xml.Node$@6040b6ef by

    (i=nmai):=(defined-piecewise(
    1wenn(n=m)
    )
    (
    (an)sonst
    )
    )
    http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdfromtonmiOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnmhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationOMFOREIGNscala.xml.Node$@6040b6ef

    The variable ii is called the index of multiplication and nn and mm the lower and upper bounds of the product respectively, together the specify the range of multiplication.

    There are variant product operators φaihttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdCollφOMFOREIGNscala.xml.Node$@6040b6ef and iSaihttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdInCollSiOMFOREIGNscala.xml.Node$@6040b6ef. The first one specifies the range of the product via a formula φφ in ii and the second one directly by giving a set SS.


  • range of summation de

    Summation is iterated addition, we define the sum over a sequnce aiOMFOREIGNscala.xml.Node$@6040b6ef by

    (i=nmai):=(defined-piecewise(
    0wenn(n=m)
    )
    (
    (an)sonst
    )
    )
    http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/numberfields/sum.omdoc?sum?sumnmiOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnmhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionOMFOREIGNscala.xml.Node$@6040b6ef

    The variable ii is called the index of summation and nn and mm the lower and upper bound of the sum respectively, together the specify the range of summation.

    There are variant summation operators (SumCollφ[i]ai)http://mathhub.info/smglom/numberfields/sum.omdoc?sum?SumCollφiOMFOREIGNscala.xml.Node$@6040b6ef and iSaihttp://mathhub.info/smglom/numberfields/sum.omdoc?sum?SumInCollSiOMFOREIGNscala.xml.Node$@6040b6ef. The first one specifies the range of the summation via a formula φφ in ii and the second one directly by giving a set SS.


  • ray Show Notations

    For distinct points AA and BB, we call the set

    (fundefeq[AB]AB)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqABhttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?rayAB

    the ray from AA through BB. We call AA the end point of ABhttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?rayAB.


  • real function

    A real function, also real-valued function, ff , is a function ff whose values fxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationfx are real numbers: (fx)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationfxhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number.


  • real number Show Notations de

    The set http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number of real numbers is defined as the completion of http://mathhub.info/smglom/numberfields/rationalnumbers.omdoc?rationalnumbers?rational-numbers.

    We use http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?negative-real-number and http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?positive-real-number for the sets of negative real numbers and positive real numbers.


  • real-valued function

    A real function, also real-valued function, ff , is a function ff whose values fxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationfx are real numbers: (fx)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationfxhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number.


  • rectangular number Show Notations de

    A pronic number Pnhttp://mathhub.info/smglom/numbers/pronicnumber.omdoc?pronicnumber?pronic-numbern, oblong number, rectangular number or heteromecic number, is a number which is the product of two consecutive natural numbers.

    (multi-relation-expressionPnequal(nn+1)equal(n2)+n)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numbers/pronicnumber.omdoc?pronicnumber?pronic-numbernhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationnn

  • reflexive de

    A relation R(A×A)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetRhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAA is called

    • reflexive on AA, iff (a,a)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairaaR for all aAhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaA, and

    • irreflexive (or anti-reflexive) on AA, iff (a,a)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?ninsethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairaaR for all aAhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaA.


  • regular de

    If in a graph GG all the vertices have the same degree, say kk, then GG is called kk-regular, or simply regular.

    A 3-regular graph is called cubic graph.


  • Regular number de

    Regular numbers are numbers that evenly divide powers of 60.


  • relation de

    R(A×B)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetRhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAB is a (binary) relation between AA and BB.

    If A=Bhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalAB then RR is called a relation on AA.


  • relation on de

    R(A×B)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetRhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAB is a (binary) relation between AA and BB.

    If A=Bhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalAB then RR is called a relation on AA.


  • relatively prime Show Notations de

    Two integers are said to be coprime (also spelled co-prime), relatively prime or mutually prime if their greatest common divisor is 1.


  • remainder

    The integer division operator computes the integer quotient (or modulus) ndivmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divnm of two natural numbers. ndivmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divnm is defined as that qhttp://mathhub.info/smglom/sets/set.omdoc?set?insetqhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, such that n=((mq)+r)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationmqr for some 0r<mhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?betweenesrm. The number rr is called the remainder and is written as nmodmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?modnm.


  • repdigit de

    A repdigit is a number that contains only the same digit.

    The representation of the repdigits in base bb is (m((bn)-1)(b-1))mhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationbnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionb for b2http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methanb n1http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methann, and (multi-relation-expression0lethanmlethanb)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanmhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanb.


  • repeated digital sum

    The number of times the digits must be summed to reach the digital sum is called a number’s additive persistence.


  • repunit Show Notations de

    A repunit is a natural number that contains only the digit 1.

    The base-b repunits are defined as (Rn(b))=(((bn)-1)(b-1))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/repunit.omdoc?repunit?repunitbnbhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationbnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionb for b2http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methanb and n1http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methann.

    A repunit prime is a repunit that is also a prime number.


  • repunit prime de

    A repunit is a natural number that contains only the digit 1.

    The base-b repunits are defined as (Rn(b))=(((bn)-1)(b-1))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/repunit.omdoc?repunit?repunitbnbhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationbnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionb for b2http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methanb and n1http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methann.

    A repunit prime is a repunit that is also a prime number.


  • restriction Show Notations de

    Let f:ABhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfAB be a function and CAhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetCA, then we call the function (fundefeq[fC]f|C)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfChttp://mathhub.info/smglom/sets/function-restriction.omdoc?function-restriction?restrictionfC the restriction of ff to CC.


  • reversal de

    The reversal of a decimal number abchttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationabc is cbahttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationcba.


  • Riemann hypothesis de

    The Riemann hypothesis is the conjecture that the nontrivial zeros of the Riemann zeta function all have real part 12http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?division.


  • Riemann integral Show Notations

    The Riemann integral of a function ff over the interval [a,b]http://mathhub.info/smglom/calculus/interval.omdoc?interval?ccintervalab is the limit of the Riemann sums.


  • Riesel number de

    A Riesel number is an odd natural number kk such that (k2n)-1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionkhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn is composite, for all natural numbers nn.


  • right coset Show Notations

    Given an element gg of the group Ghttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureG and one of its subgroups HH, we define the left coset (respectively the right coset) of HH with gg as (fundefeq[gH]gH)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqgHhttp://mathhub.info/smglom/algebra/coset.omdoc?coset?left-cosetgH (respectively (fundefeq[gH]Hg)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqgHhttp://mathhub.info/smglom/algebra/coset.omdoc?coset?right-cosetgH).


  • right divides de

    Let Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR be a ring and m0http://mathhub.info/smglom/mv/equal.omdoc?equal?notequalm, then we say that mm is a left divisor of nn (or mm left divides nn) and that nn is a left multiple of mm, if there is a kRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetkR such that (multiplication)=nhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationn.

    Analogously we say that mm is a right divisor of nn (or mm right divides nn) and that nn is a right multiple of mm, if there is a kRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetkR such that (multiplication)=nhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationn.

    If RR is commxutative, then left and right divisors coincide and we simply speak of divisor and multiple and write m|nhttp://mathhub.info/smglom/algebra/ring-divisor.omdoc?ring-divisor?divisormn.

    1, 1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction, nn, and nhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn are known as the trivial divisors of nn. A divisor of nn that is not a trivial divisor is known as a proper or non-trivial divisor.


  • right divisor de

    Let Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR be a ring and m0http://mathhub.info/smglom/mv/equal.omdoc?equal?notequalm, then we say that mm is a left divisor of nn (or mm left divides nn) and that nn is a left multiple of mm, if there is a kRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetkR such that (multiplication)=nhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationn.

    Analogously we say that mm is a right divisor of nn (or mm right divides nn) and that nn is a right multiple of mm, if there is a kRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetkR such that (multiplication)=nhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationn.

    If RR is commxutative, then left and right divisors coincide and we simply speak of divisor and multiple and write m|nhttp://mathhub.info/smglom/algebra/ring-divisor.omdoc?ring-divisor?divisormn.

    1, 1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction, nn, and nhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn are known as the trivial divisors of nn. A divisor of nn that is not a trivial divisor is known as a proper or non-trivial divisor.


  • right multiple de

    Let Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR be a ring and m0http://mathhub.info/smglom/mv/equal.omdoc?equal?notequalm, then we say that mm is a left divisor of nn (or mm left divides nn) and that nn is a left multiple of mm, if there is a kRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetkR such that (multiplication)=nhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationn.

    Analogously we say that mm is a right divisor of nn (or mm right divides nn) and that nn is a right multiple of mm, if there is a kRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetkR such that (multiplication)=nhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationn.

    If RR is commxutative, then left and right divisors coincide and we simply speak of divisor and multiple and write m|nhttp://mathhub.info/smglom/algebra/ring-divisor.omdoc?ring-divisor?divisormn.

    1, 1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction, nn, and nhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn are known as the trivial divisors of nn. A divisor of nn that is not a trivial divisor is known as a proper or non-trivial divisor.


  • right-truncatable prime de

    A two-sided prime is both left-truncatable and right-truncatable.

    For example: 3137

    A right-truncatable prime is a prime number which remains prime when the last (“right”) digit is successively removed.

    For example: 3793


  • rightsided limit Show Notations de

    Let f:SThttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfST with S,Thttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?msseteqSThttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number and ahttp://mathhub.info/smglom/sets/set.omdoc?set?insetahttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number.

    Then the leftsided limit of ff at aa (also: the limit of ff as xx approaches aa from below) is defined to be the lhttp://mathhub.info/smglom/sets/set.omdoc?set?insetlhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number, sucht that for every ϵ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanϵ there is a δ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanδ such that (|(((f))l)|)ϵhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/numberfields/absolutevalue.omdoc?absolutevalue?absolute-valuehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionflϵ whenever (multi-relation-expression0lethan(a)xlethanδ)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionaxhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanδ. The leftsided limit is written as limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?leftsided-limitaxf, limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?leftsided-limitaxf, or limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?leftsided-limitaxf.

    Analogously, the rightsided limit at aa (also: the limit of ff as xx approaches aa from above) is the lhttp://mathhub.info/smglom/sets/set.omdoc?set?insetlhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number, such that for every ϵ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanϵ there is a δ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanδ such that (|(((f))l)|)ϵhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/numberfields/absolutevalue.omdoc?absolutevalue?absolute-valuehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionflϵ whenever (multi-relation-expression0lethan(x)alethanδ)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionxahttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanδ. The rightsided limit is written as limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?rightsided-limitaxf, limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?rightsided-limitaxf, or limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?rightsided-limitaxf.


  • ring de

    A structure Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR is called a ring, if Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR is a Abelian group (called the additive structure), Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR is a monoid (called the multiplicative structure), and Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR is a ringoid.

    We call 0 the zero of the ring and 1 the one of the ring.


  • ring de

    A ring is a semiring Shttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureS such that Shttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureS forms an Abelian group.


  • root

    A tree is a directed acyclic graph G=(V,E)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalGhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE such that

    • there is exactly one initialnode vrVhttp://mathhub.info/smglom/sets/set.omdoc?set?insetOMFOREIGNscala.xml.Node$@6040b6efV (called the root), and

    • all nodes but the root have indegree 1.

    We call vv the parent of ww, iff (v,w)Ehttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairvwE (ww is a child of vv). We call a node vv a leaf of GG, iff it is terminal, i.e. if it does not have children.


  • root mean square Show Notations de

    The root mean square of a set {(x1,,xn)}http://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlixn is given by:

    =(((1n)((x12)+(x22)++(xn2))))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/smglom/rootmeansquare.omdoc?rootmeansquare?root-mean-squarehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?square-roothttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationOMFOREIGNscala.xml.Node$@6040b6ef

  • safe prime de

    A safe prime pp is a prime number where (p-1)2http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionp is also a prime.


  • Sarrus number de

    Sarrus numbers are Fermat pseudoprimes to base 2.


  • Schröder number de

    A Schröder number is the number of paths from the southwest corner 0,0http://mathhub.info/smglom/sets/pair.omdoc?pair?pair of an nnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnn grid to the northeast corner n,nhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairnn, using only single steps north, northeast, or east, that do not rise above the SW-NE diagonal.

    The first few Schröder numbers are 1,2,6,22,90,394,1806,8558,...http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?infseq


  • Schröder-Hipparchus number de

    The Schröder-Hipparchus number (or super-Catalan number) S(n)http://mathhub.info/smglom/numbers/schroederhipparchusnumberrec.omdoc?schroederhipparchusnumberrec?schroederhipprecn is for natural numbers nn defind by the recurrence relation:

    S(1)http://mathhub.info/smglom/numbers/schroederhipparchusnumberrec.omdoc?schroederhipparchusnumberrec?schroederhipprec equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal (S(2))=1http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/schroederhipparchusnumberrec.omdoc?schroederhipparchusnumberrec?schroederhipprec
    S(n)http://mathhub.info/smglom/numbers/schroederhipparchusnumberrec.omdoc?schroederhipparchusnumberrec?schroederhipprecn equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal (1n)((((6n)-9)(S((n-1))))-((n-3)(S((n-2)))))http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnhttp://mathhub.info/smglom/numbers/schroederhipparchusnumberrec.omdoc?schroederhipparchusnumberrec?schroederhipprechttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numbers/schroederhipparchusnumberrec.omdoc?schroederhipparchusnumberrec?schroederhipprechttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn

  • Schröder-Hipparchus number Show Notations de

    The Schröder-Hipparchus number (or super-Catalan number) S(n)http://mathhub.info/smglom/numbers/schroederhipparchusnumberformula.omdoc?schroederhipparchusnumberformula?Schroeder-Hipparchus-numbern is for natural numbers nn defined by

    S(n)http://mathhub.info/smglom/numbers/schroederhipparchusnumberformula.omdoc?schroederhipparchusnumberformula?Schroeder-Hipparchus-numbern

  • Schröder-Hipparchus number Show Notations

    The Schröder-Hipparchus number (or super-Catalan number) S(n)http://mathhub.info/smglom/numbers/schroederhipparchusnumber.omdoc?schroederhipparchusnumber?Schroeder-Hipparchus-numbern is the number of plane trees with nn leaves.

    The first Schröder-Hipparchus numbers are: 1,1,3,11,45,197,903,4279,20793,103049,...http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?infseq


  • scientific notation Show Notations

    In scientific notation all numbers are written in the form of a×10bhttp://mathhub.info/smglom/numberfields/scinotation.omdoc?scinotation?scientific-notationab, (ahttp://mathhub.info/smglom/sets/set.omdoc?set?insetahttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number times 10 raised to the power of bhttp://mathhub.info/smglom/sets/set.omdoc?set?insetbhttp://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?integers), aa is called the significand or mantissa.

    a×10bhttp://mathhub.info/smglom/numberfields/scinotation.omdoc?scinotation?scientific-notationab is called normalized, iff 1(|a|)<10http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?betweeneshttp://mathhub.info/smglom/numberfields/absolutevalue.omdoc?absolutevalue?absolute-valuea


  • second Chebyshev de

  • second derivative de

    For any nhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number we define the nnth derivative of a function f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN as

    (fundefeq[nfxa]dnfdxn|x=a)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqnfxahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa

    The first derivative d1dx1fhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtfxof ff is Dxfhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfx, d2dx2fhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtfx is the second derivative of ff, d3dx3fhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtfx the third derivative of ff, etc. In Leibniz’ notation the nnth derivative function of ff is denoted by dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx.

    We call ff nn times differentiable at aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM, iff dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa exists as a limit. Analogously, ff is nn times differentiable on MM, iff dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx exists and is total on MM.

    ff is called infinitely differentiable or smooth at aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM or on MM, iff dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa and dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx exist for all nhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number respectively.


  • second Skewes number Show Notations de

    The second Skewes number http://mathhub.info/smglom/smglom/skewesnumber.omdoc?skewesnumber?second-Skewes-number is the number above which (π(n))(Li(n))http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?NumberPrimeNumbernhttp://mathhub.info/smglom/smglom/logarithmicintegralbig.omdoc?logarithmicintegralbig?logarithmicintbign must fail assuming that the Riemann hypothesis is false.

    =(10(10(101000)))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/smglom/skewesnumber.omdoc?skewesnumber?second-Skewes-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiation

  • second smarandache constant Show Notations de

    The second smarandache constant http://mathhub.info/smglom/smglom/smarandacheconstant2.omdoc?smarandacheconstant2?second-smarandache-constant is defind by

    http://mathhub.info/smglom/smglom/smarandacheconstant2.omdoc?smarandacheconstant2?second-smarandache-constant

    where S(n)http://mathhub.info/smglom/smglom/smarandachefunction.omdoc?smarandachefunction?smarandache-functionn is the smarandache function.


  • segment Show Notations

    For distinct points AA and BB, we call the set

    (fundefeq[AB]AB)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqABhttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?segmentAB

    the segment between AA and BB.


  • self prime de

  • semiprime de

  • semiring de

    A semiring is a ringoid Shttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureS such that Shttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureS and Shttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureS are monoids and additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?addition is commutative.


  • sequence of partial sums de

    For any sequence (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan with (an)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselanhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number or (an)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselanhttp://mathhub.info/smglom/numberfields/complexnumbers.omdoc?complexnumbers?complex-number we define the nn-th partial sum (fundefeq[an]sn)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqanhttp://mathhub.info/smglom/calculus/series.omdoc?series?partial-suman.

    The series induced by (na)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequencean is the sequence of partial sums (sequenceon[an]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonann.


  • sequence of partial sums de

    For any sequence (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan with (an)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselanhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number or (an)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselanhttp://mathhub.info/smglom/numberfields/complexnumbers.omdoc?complexnumbers?complex-number we define the nn-th partial sum (fundefeq[an]sn)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqanhttp://mathhub.info/smglom/calculus/series.omdoc?series?partial-suman.

    The series induced by (na)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequencean is the sequence of partial sums (sequenceon[an]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonann.


  • series de

    For any sequence (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan with (an)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselanhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number or (an)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselanhttp://mathhub.info/smglom/numberfields/complexnumbers.omdoc?complexnumbers?complex-number we define the nn-th partial sum (fundefeq[an]sn)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqanhttp://mathhub.info/smglom/calculus/series.omdoc?series?partial-suman.

    The series induced by (na)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequencean is the sequence of partial sums (sequenceon[an]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonann.


  • seventh smarandache constant Show Notations de

    The seventh smarandache constant http://mathhub.info/smglom/smglom/smarandacheconstant7.omdoc?smarandacheconstant7?seventh-smarandache-constant is defind for a natural number kk by

    http://mathhub.info/smglom/smglom/smarandacheconstant7.omdoc?smarandacheconstant7?seventh-smarandache-constant

    where S(n)http://mathhub.info/smglom/smglom/smarandachefunction.omdoc?smarandachefunction?smarandache-functionn is the smarandache function.

    The sum converges.


  • Sexy primes de

    Sexy primes are prime numbers that differ from each other by six.


  • Sierpinski number de

    A Sierpinski number is an odd natural numbers kk such that (k2n)+1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionkhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn is composite, for all natural numbers nn.


  • significand

    In scientific notation all numbers are written in the form of a×10bhttp://mathhub.info/smglom/numberfields/scinotation.omdoc?scinotation?scientific-notationab, (ahttp://mathhub.info/smglom/sets/set.omdoc?set?insetahttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number times 10 raised to the power of bhttp://mathhub.info/smglom/sets/set.omdoc?set?insetbhttp://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?integers), aa is called the significand or mantissa.

    a×10bhttp://mathhub.info/smglom/numberfields/scinotation.omdoc?scinotation?scientific-notationab is called normalized, iff 1(|a|)<10http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?betweeneshttp://mathhub.info/smglom/numberfields/absolutevalue.omdoc?absolutevalue?absolute-valuea


  • simple de

    Given a directed graph G=(V,E)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalGhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE,

    • a path pp is called cyclic (or a cycle) iff (start(p))=(end(p))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pstartphttp://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pendp.

    • a cycle (v0,,vn)http://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlivn is called simple, iff vivjhttp://mathhub.info/smglom/mv/equal.omdoc?equal?notequalOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6ef for i,j1nhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?mbetweeneeijn with ijhttp://mathhub.info/smglom/mv/equal.omdoc?equal?notequalij.

    • GG is called acyclic (or a DAG (directed acyclic graph)) iff there is no cycle in GG.


  • simple order de

    We call a partial ordering RR a total ordering (or simple order or linear order), iff abhttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?poleab or bahttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?poleba for all a,bAhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabA.

    We call a structure S,polehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureShttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole of a set SS and a total ordering polehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole an totally ordered set.


  • simply normal

    A real numbers xx is said to be simply normal if its digits in every base are equally likely.

    A number is absolutely non-normal or absolutely abnormal if it is not simply normal in any base.

    xx is said to be simply normal in base bb if its digits in base bb are equally likely.


  • simply normal in base de

    A number is absolutely non-normal or absolutely abnormal if it is not simply normal in any base.

    xx is said to be simply normal in base bb if its digits in base bb are equally likely.


  • sixteenth smarandache constant de

    The sixteenth smarandache constant sixteenth-smarandache-constanthttp://mathhub.info/smglom/smglom/smarandacheconstant16.omdoc?smarandacheconstant16?sixteenth-smarandache-constant is defind by

    s16(α)http://mathhub.info/smglom/smglom/smarandacheconstant16.omdoc?smarandacheconstant16?sixteenth-smarandache-constantα

    where S(n)http://mathhub.info/smglom/smglom/smarandachefunction.omdoc?smarandachefunction?smarandache-functionn is the smarandache function.

    The sum converges for all real numbers α>1http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanα.


  • sixth smarandache constant Show Notations de

    The sixth smarandache constant http://mathhub.info/smglom/smglom/smarandacheconstant6.omdoc?smarandacheconstant6?sixth-smarandache-constant is defind by

    http://mathhub.info/smglom/smglom/smarandacheconstant6.omdoc?smarandacheconstant6?sixth-smarandache-constant

    where S(n)http://mathhub.info/smglom/smglom/smarandachefunction.omdoc?smarandachefunction?smarandache-functionn is the smarandache function.

    The sum converges.

    (multi-relation-expression0.218282lethanlethan0.5)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/smglom/smarandacheconstant6.omdoc?smarandacheconstant6?sixth-smarandache-constanthttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethan

    .


  • size Show Notations de

    We say that a set AA is finite and has cardinality (or size) (#(A))http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityAhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, iff there is a bijective function f:A({n|(n<(#(A)))})http://mathhub.info/smglom/sets/functions.omdoc?functions?funfAhttp://mathhub.info/smglom/sets/set.omdoc?set?rsetsthttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-numbernhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthannhttp://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityA.

    The cardinality of a set AA is also written as #(A)http://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityA, #(A)http://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityA, #(A)http://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityA, or #(A)http://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityA.


  • Skewes number Show Notations de

    The second Skewes number http://mathhub.info/smglom/smglom/skewesnumber.omdoc?skewesnumber?second-Skewes-number is the number above which (π(n))(Li(n))http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?NumberPrimeNumbernhttp://mathhub.info/smglom/smglom/logarithmicintegralbig.omdoc?logarithmicintegralbig?logarithmicintbign must fail assuming that the Riemann hypothesis is false.

    =(10(10(101000)))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/smglom/skewesnumber.omdoc?skewesnumber?second-Skewes-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiation

  • small set de

    A small set of positive integers

    S=({(s0,s1,s2,s3,)})http://mathhub.info/smglom/mv/equal.omdoc?equal?equalShttp://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?aseqdotsOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6ef

    is a set for that the infinite sum

    (1s0)+(1s1)+(1s2)+(1s3)+http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionOMFOREIGNscala.xml.Node$@6040b6ef

    converges.


  • smallest common multiple Show Notations de

    The least common multiple (lcm) (also called the lowest common multiple or smallest common multiple) of two or more integers is the smallest positive integer that is divisible by all given integers.


  • Smarandache function de

    The Smarandache function S(n)http://mathhub.info/smglom/smglom/smarandachefunction.omdoc?smarandachefunction?smarandache-functionn is defined for a given positive integer nn to be the smallest number such that nn divides its factorial.

    S(n)http://mathhub.info/smglom/smglom/smarandachefunction.omdoc?smarandachefunction?smarandache-functionn

  • smooth de

    We call ff nn times differentiable at aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM, iff dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa exists as a limit. Analogously, ff is nn times differentiable on MM, iff dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx exists and is total on MM.

    ff is called infinitely differentiable or smooth at aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM or on MM, iff dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa and dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx exist for all nhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number respectively.


  • Sophie Germain prime de

    A prime number pp is a Sophie Germain prime if (2p)+1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationp is also prime.


  • sphenic number de

    A sphenic number is a natural number that is the product of three distinct prime numbers.


  • square-full number de

    A powerful number is a positive integer mm such that for every prime number pp dividing mm, p2http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationp also divides mm.

    A powerful number is the product of a square and a cube. Powerful numbers are also known as squareful numbers, square-full numbers, or 2-full numbers.


  • squareful number de

    A powerful number is a positive integer mm such that for every prime number pp dividing mm, p2http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationp also divides mm.

    A powerful number is the product of a square and a cube. Powerful numbers are also known as squareful numbers, square-full numbers, or 2-full numbers.


  • start de

    Given a directed graph G=(V,E)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalGhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE we call a vector (p=((v0,,vn)))(V(n+1))http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalphttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlivnhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?nCartSpaceVhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionn a path in GG iff (vi-1,vi)Ehttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6efE for all 1inhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?betweeneein, n>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethann.

    • v0OMFOREIGNscala.xml.Node$@6040b6ef is called the start of pp (write start(p)http://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pstartp )

    • vnOMFOREIGNscala.xml.Node$@6040b6ef is called the end of pp (write end(p)http://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pendp )

    • nn is called the length of pp (write len(p)http://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?plenp )


  • Stern prime de

    If for a prime qq there is no smaller prime pp and nonzero integer bb such that

    q=(p+(2(b2)))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalqhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationb

    then qq is a Stern prime.


  • Stoneham number de

    For coprime numbers (morethanbc1)http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanbc, the Stoneham number αb,chttp://mathhub.info/smglom/smglom/stonehamnumber.omdoc?stonehamnumber?stonehambc is defined as

    αb,chttp://mathhub.info/smglom/smglom/stonehamnumber.omdoc?stonehamnumber?stonehambc

  • strict ordering de

    We call a partial ordering RR a total ordering (or simple order or linear order), iff abhttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?poleab or bahttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?poleba for all a,bAhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabA.

    We call a structure S,polehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureShttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole of a set SS and a total ordering polehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole an totally ordered set.

    We call a preorder pole(A×A)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsethttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?polehttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAA on AA a partial ordering (or partial order), iff it is antisymmetric. We associate with polehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole a strict ordering poless:=(bsetst[ab](a,b)pole)http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?polesshttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstabhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairabhttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole.

    We often also use the converse relations pomehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pome and pomorehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pomore.

    We call a structure S,polehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureShttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole of a set SS and a partial ordering polehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole an ordered set.


  • strong prime de

    A strong prime is a prime number that is greater than the arithmetic mean of the nearest primes above and below.

    (pn)>(((p(n-1))+(p(n+1)))2)http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanhttp://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?prime-numbernhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?prime-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?prime-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionn

    where pnhttp://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?prime-numbern is the nnth prime number.


  • structure Show Notations de

    A structure combines multiple mathematical objects (the components) into a new object. Structures are usually given as finite enumerations, where the components have names by which they can be referenced.


  • subcover de

    A cover of a set XX is a collection CC of sets, such that X(aAUa)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetXhttp://mathhub.info/smglom/sets/union.omdoc?union?munionCollectionAaOMFOREIGNscala.xml.Node$@6040b6ef. A subset CChttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efC is called a subcover of XX, iff it still covers XX.


  • subharmonic series de

    Let Mhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetMhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, then an infinite series nM(1n)http://mathhub.info/smglom/numberfields/sum.omdoc?sum?SumInCollMnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionn is called a subharmonic series.


  • subset Show Notations ro de tr

    A set AA is a subset of a set BB (written ABhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?proper-subsetAB), iff all xAhttp://mathhub.info/smglom/sets/set.omdoc?set?insetxA are members of BB.


  • sum Show Notations de

    Summation is iterated addition, we define the sum over a sequnce aiOMFOREIGNscala.xml.Node$@6040b6ef by

    (i=nmai):=(defined-piecewise(
    0wenn(n=m)
    )
    (
    (an)sonst
    )
    )
    http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/numberfields/sum.omdoc?sum?sumnmiOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnmhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionOMFOREIGNscala.xml.Node$@6040b6ef

    The variable ii is called the index of summation and nn and mm the lower and upper bound of the sum respectively, together the specify the range of summation.

    There are variant summation operators (SumCollφ[i]ai)http://mathhub.info/smglom/numberfields/sum.omdoc?sum?SumCollφiOMFOREIGNscala.xml.Node$@6040b6ef and iSaihttp://mathhub.info/smglom/numberfields/sum.omdoc?sum?SumInCollSiOMFOREIGNscala.xml.Node$@6040b6ef. The first one specifies the range of the summation via a formula φφ in ii and the second one directly by giving a set SS.


  • sum of squares function Show Notations de

    The function rn(k)http://mathhub.info/smglom/smglom/sumofsquaresfunction.omdoc?sumofsquaresfunction?sumof-squares-functionnk is called sum of squares function.

    (rn(k))=(#((bsetst[an](a1,a2,,an)(n))))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/smglom/sumofsquaresfunction.omdoc?sumofsquaresfunction?sumof-squares-functionnkhttp://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstanhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tupleOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?nCartSpacehttp://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?integersn

  • sum-of-divisors function Show Notations de

    The sum-of-divisors function σ1(n)http://mathhub.info/smglom/smglom/sumofdivisorsfunction.omdoc?sumofdivisorsfunction?sumdiv-functionn (also written as σ1(n)http://mathhub.info/smglom/smglom/sumofdivisorsfunction.omdoc?sumofdivisorsfunction?sumdiv-functionn) is defined as the sum of the positive divisors of nn, i.e. σ1(n)http://mathhub.info/smglom/smglom/sumofdivisorsfunction.omdoc?sumofdivisorsfunction?sumdiv-functionn


  • summatory von Mangoldt function de

    The summatory von Mangoldt function, also known as the Chebyshev function, is defined as

    ψψ

  • super-Catalan number de

    The Schröder-Hipparchus number (or super-Catalan number) S(n)http://mathhub.info/smglom/numbers/schroederhipparchusnumberrec.omdoc?schroederhipparchusnumberrec?schroederhipprecn is for natural numbers nn defind by the recurrence relation:

    S(1)http://mathhub.info/smglom/numbers/schroederhipparchusnumberrec.omdoc?schroederhipparchusnumberrec?schroederhipprec equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal (S(2))=1http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/schroederhipparchusnumberrec.omdoc?schroederhipparchusnumberrec?schroederhipprec
    S(n)http://mathhub.info/smglom/numbers/schroederhipparchusnumberrec.omdoc?schroederhipparchusnumberrec?schroederhipprecn equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal (1n)((((6n)-9)(S((n-1))))-((n-3)(S((n-2)))))http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnhttp://mathhub.info/smglom/numbers/schroederhipparchusnumberrec.omdoc?schroederhipparchusnumberrec?schroederhipprechttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numbers/schroederhipparchusnumberrec.omdoc?schroederhipparchusnumberrec?schroederhipprechttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn

  • super-Catalan number Show Notations de

    The Schröder-Hipparchus number (or super-Catalan number) S(n)http://mathhub.info/smglom/numbers/schroederhipparchusnumberformula.omdoc?schroederhipparchusnumberformula?Schroeder-Hipparchus-numbern is for natural numbers nn defined by

    S(n)http://mathhub.info/smglom/numbers/schroederhipparchusnumberformula.omdoc?schroederhipparchusnumberformula?Schroeder-Hipparchus-numbern

  • super-Catalan number Show Notations

    The Schröder-Hipparchus number (or super-Catalan number) S(n)http://mathhub.info/smglom/numbers/schroederhipparchusnumber.omdoc?schroederhipparchusnumber?Schroeder-Hipparchus-numbern is the number of plane trees with nn leaves.

    The first Schröder-Hipparchus numbers are: 1,1,3,11,45,197,903,4279,20793,103049,...http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?infseq


  • superior limit Show Notations

    Ist M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd ein metrischer Raum und (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan eine Folge mit (ai)Mhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselaiM für ihttp://mathhub.info/smglom/sets/set.omdoc?set?insetihttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, so ist der

    • Limes superior limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (schreibe auch limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) definiert durch

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai
    • Limes inferior limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (schreibe auch limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) definiert durch

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai

    Let M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd be a metric space and (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan a sequence with (ai)Mhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselaiM for all ihttp://mathhub.info/smglom/sets/set.omdoc?set?insetihttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, then the

    • limit superior (also called supremum limit, superior limit, upper limit, or outer limit) limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (also written as limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) is defined as

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai
    • limit inferior (also called infimum limit, inferior limit, lower limit, or inner limit) limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (also written as limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) is defined as

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai

  • superset Show Notations ro de tr

    A set AA is a superset of a set BB (written ABhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?proper-supersetAB), iff BAhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?proper-subsetBA.


  • supremum Show Notations de

    Let S,polehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureShttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole be an ordered set and TShttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetTS, then we call the smallest upper bound sup(T)http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?supremumT (largest lower bound inf(T)http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?infimumT) of TT the supremum or least upper bound (infimum or greatest lower bound) of TT (if it exists).

    If ee is an expression and φφ a condition (in a variable xx), we write (bsupremumφ[x]e)http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?bsupremumφxe for sup((bsetst[x]e))http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?supremumhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstxe and call it the supremum for ee over φφ. Analogously, we write (binfimumφ[x]e)http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?binfimumφxe for inf((bsetst[x]φ))http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?infimumhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstxφ and call it the infimum for ee over φφ


  • supremum limit Show Notations

    Ist M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd ein metrischer Raum und (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan eine Folge mit (ai)Mhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselaiM für ihttp://mathhub.info/smglom/sets/set.omdoc?set?insetihttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, so ist der

    • Limes superior limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (schreibe auch limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) definiert durch

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai
    • Limes inferior limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (schreibe auch limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) definiert durch

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai

    Let M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd be a metric space and (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan a sequence with (ai)Mhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselaiM for all ihttp://mathhub.info/smglom/sets/set.omdoc?set?insetihttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, then the

    • limit superior (also called supremum limit, superior limit, upper limit, or outer limit) limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (also written as limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) is defined as

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai
    • limit inferior (also called infimum limit, inferior limit, lower limit, or inner limit) limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (also written as limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) is defined as

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai

  • surjective de

    A function f:SThttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfST is called surjective iff for all yThttp://mathhub.info/smglom/sets/set.omdoc?set?insetyT there is a xShttp://mathhub.info/smglom/sets/set.omdoc?set?insetxS with (f)=yhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalfy.


  • symmetric de

    A relation R(A×A)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetRhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAA is called

    • symmetric on AA, iff (b,a)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairbaR for all a,bAhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabA with (a,b)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairabR.

    • asymmetric on AA, iff (b,a)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?ninsethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairbaR for all a,bAhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabA with (a,b)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairabR.

    • antisymmetric on AA, iff (a,b)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairabR and (b,a)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairbaR imply a=bhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalab.


  • symmetry de

    Let MM be a set, then we call a function d:(M×M)http://mathhub.info/smglom/sets/functions.omdoc?functions?fundhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsMMhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number a distance function (or metric ) on MM, iff for all x,y,zMhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetxyzM the following three identities hold:

    1. 1.

      (d)=0http://mathhub.info/smglom/mv/equal.omdoc?equal?equald iff x=yhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalxy (identity of indiscernibles),

    2. 2.

      (d)=(d)http://mathhub.info/smglom/mv/equal.omdoc?equal?equaldd (symmetry), and

    3. 3.

      (d)((d)+(d))http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethandhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additiondd (triangle inequality).

    We call M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd a metric space with base set MM and metric dd.


  • taxicab number Show Notations de

    The nnth taxicab number Ta(n)http://mathhub.info/smglom/numbers/taxicabnumber.omdoc?taxicabnumber?taxicab-numbern is defined as the smallest number that can be expressed as a sum of two positive cubes in nn distinct ways.

    For example:

    (multi-relation-expressionTa(3)equal87539319equal(1673)+(4363))http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numbers/taxicabnumber.omdoc?taxicabnumber?taxicab-numberhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiation
    equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal
    equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal

  • Taylor series Show Notations de

    Let ff be a real-or complex-valued function that is smooth at a limit point aa of the domain of ff, then we call the infinite series given by

    (fundefeq[fxa]𝑇f(x;a))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfxahttp://mathhub.info/smglom/calculus/Taylor-series.omdoc?Taylor-series?Taylor-seriesfxa

    the Taylor series for ff around aa. If a=0http://mathhub.info/smglom/mv/equal.omdoc?equal?equala, then the series is known as the Mclaurin series.


  • tenth smarandache constants

    The tenth smarandache constants http://mathhub.info/smglom/smglom/smarandacheconstant10.omdoc?smarandacheconstant10?tenth-smarandache-constant are defind by

    s10(α)http://mathhub.info/smglom/smglom/smarandacheconstant10.omdoc?smarandacheconstant10?smarandacheconsttenα

    where S(n)http://mathhub.info/smglom/smglom/smarandachefunction.omdoc?smarandachefunction?smarandache-functionn is the smarandache function.

    The sum converges for all real number α>1http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanα.


  • th derivative de

    For any nhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number we define the nnth derivative of a function f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN as

    (fundefeq[nfxa]dnfdxn|x=a)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqnfxahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa

    The first derivative d1dx1fhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtfxof ff is Dxfhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfx, d2dx2fhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtfx is the second derivative of ff, d3dx3fhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtfx the third derivative of ff, etc. In Leibniz’ notation the nnth derivative function of ff is denoted by dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx.

    We call ff nn times differentiable at aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM, iff dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa exists as a limit. Analogously, ff is nn times differentiable on MM, iff dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx exists and is total on MM.

    ff is called infinitely differentiable or smooth at aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM or on MM, iff dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa and dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx exist for all nhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number respectively.


  • Thabit number de

    A Thabit number is an integer of the form

    (32n)-1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn

    for a non-negative integer nn.


  • third derivative de

    For any nhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number we define the nnth derivative of a function f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN as

    (fundefeq[nfxa]dnfdxn|x=a)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqnfxahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa

    The first derivative d1dx1fhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtfxof ff is Dxfhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfx, d2dx2fhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtfx is the second derivative of ff, d3dx3fhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtfx the third derivative of ff, etc. In Leibniz’ notation the nnth derivative function of ff is denoted by dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx.

    We call ff nn times differentiable at aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM, iff dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa exists as a limit. Analogously, ff is nn times differentiable on MM, iff dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx exists and is total on MM.

    ff is called infinitely differentiable or smooth at aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM or on MM, iff dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa and dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx exist for all nhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number respectively.


  • third smarandache constant Show Notations de

    The third smarandache constant http://mathhub.info/smglom/smglom/smarandacheconstant3.omdoc?smarandacheconstant3?third-smarandache-constant is defind by

    http://mathhub.info/smglom/smglom/smarandacheconstant3.omdoc?smarandacheconstant3?third-smarandache-constant

    where S(n)http://mathhub.info/smglom/smglom/smarandachefunction.omdoc?smarandachefunction?smarandache-functionn is the smarandache function.


  • thirteenth smarandache constant Show Notations de

    The thirteenth smarandache constant http://mathhub.info/smglom/smglom/smarandacheconstant13.omdoc?smarandacheconstant13?thirteenth-smarandache-constant is defind by

    http://mathhub.info/smglom/smglom/smarandacheconstant13.omdoc?smarandacheconstant13?thirteenth-smarandache-constant

    where S(n)http://mathhub.info/smglom/smglom/smarandachefunction.omdoc?smarandachefunction?smarandache-functionn is the smarandache function.

    The sum converges.


  • times differentiable de

    We call ff nn times differentiable at aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM, iff dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa exists as a limit. Analogously, ff is nn times differentiable on MM, iff dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx exists and is total on MM.

    ff is called infinitely differentiable or smooth at aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM or on MM, iff dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa and dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx exist for all nhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number respectively.


  • topological space de

    A topological space X,Ohttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureXOMFOREIGNscala.xml.Node$@6040b6ef is a set XX together with a collection O(𝒫(X))http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/sets/powerset.omdoc?powerset?powersetX, such that

    1. 1.

      ,XOhttp://mathhub.info/smglom/sets/set.omdoc?set?minsethttp://mathhub.info/smglom/sets/emptyset.omdoc?emptyset?esetXOMFOREIGNscala.xml.Node$@6040b6ef.

    2. 2.

      (SSs)Ohttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/union.omdoc?union?munionCollectionOMFOREIGNscala.xml.Node$@6040b6efSsOMFOREIGNscala.xml.Node$@6040b6ef if SShttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetSOMFOREIGNscala.xml.Node$@6040b6ef.

    3. 3.

      (SSs)Ohttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/intersection.omdoc?intersection?mintersectCollectionOMFOREIGNscala.xml.Node$@6040b6efSsOMFOREIGNscala.xml.Node$@6040b6ef if SShttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetSOMFOREIGNscala.xml.Node$@6040b6ef is finite.

    OOMFOREIGNscala.xml.Node$@6040b6ef is called an open set topology (or just topology) on XX. Members of a topology OOMFOREIGNscala.xml.Node$@6040b6ef are called open sets and their complements closed sets. A subset of XX may be neither closed nor open, either closed or open, or both. A set that is both closed and open is called a clopen set.


  • topology de

    A topological space X,Ohttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureXOMFOREIGNscala.xml.Node$@6040b6ef is a set XX together with a collection O(𝒫(X))http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/sets/powerset.omdoc?powerset?powersetX, such that

    1. 1.

      ,XOhttp://mathhub.info/smglom/sets/set.omdoc?set?minsethttp://mathhub.info/smglom/sets/emptyset.omdoc?emptyset?esetXOMFOREIGNscala.xml.Node$@6040b6ef.

    2. 2.

      (SSs)Ohttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/union.omdoc?union?munionCollectionOMFOREIGNscala.xml.Node$@6040b6efSsOMFOREIGNscala.xml.Node$@6040b6ef if SShttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetSOMFOREIGNscala.xml.Node$@6040b6ef.

    3. 3.

      (SSs)Ohttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/intersection.omdoc?intersection?mintersectCollectionOMFOREIGNscala.xml.Node$@6040b6efSsOMFOREIGNscala.xml.Node$@6040b6ef if SShttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetSOMFOREIGNscala.xml.Node$@6040b6ef is finite.

    OOMFOREIGNscala.xml.Node$@6040b6ef is called an open set topology (or just topology) on XX. Members of a topology OOMFOREIGNscala.xml.Node$@6040b6ef are called open sets and their complements closed sets. A subset of XX may be neither closed nor open, either closed or open, or both. A set that is both closed and open is called a clopen set.


  • total function de

    If f:XYhttp://mathhub.info/smglom/sets/functions.omdoc?functions?partfunfXY is a total relation (i.e. for all xXhttp://mathhub.info/smglom/sets/set.omdoc?set?insetxX there is a unique yYhttp://mathhub.info/smglom/sets/set.omdoc?set?insetyY with (x,y)fhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairxyf), we call ff a total function and write f:XYhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfXY.


  • total ordering de

    We call a partial ordering RR a total ordering (or simple order or linear order), iff abhttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?poleab or bahttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?poleba for all a,bAhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabA.

    We call a structure S,polehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureShttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole of a set SS and a total ordering polehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole an totally ordered set.


  • totally ordered set de

    We call a partial ordering RR a total ordering (or simple order or linear order), iff abhttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?poleab or bahttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?poleba for all a,bAhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabA.

    We call a structure S,polehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureShttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole of a set SS and a total ordering polehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole an totally ordered set.


  • transitive de

    A relation R(A×A)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetRhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAA is called transitive on AA, iff (a,c)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairacR for all a,b,cAhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabcA with (a,b)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairabR and (b,c)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairbcR.


  • transitive-reflexive closure Show Notations de

    Let RR be a binary relation, then we call the smallest transitive, reflexive relation that contains RR the transitive-reflexive closure of RR we denote it with Rhttp://mathhub.info/smglom/sets/transitive-closure.omdoc?transitive-closure?transitive-reflexive-closureR.


  • tree de

    A tree is a directed acyclic graph G=(V,E)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalGhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE such that

    • there is exactly one initialnode vrVhttp://mathhub.info/smglom/sets/set.omdoc?set?insetOMFOREIGNscala.xml.Node$@6040b6efV (called the root), and

    • all nodes but the root have indegree 1.

    We call vv the parent of ww, iff (v,w)Ehttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairvwE (ww is a child of vv). We call a node vv a leaf of GG, iff it is terminal, i.e. if it does not have children.


  • triangle inequality de

    Let MM be a set, then we call a function d:(M×M)http://mathhub.info/smglom/sets/functions.omdoc?functions?fundhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsMMhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number a distance function (or metric ) on MM, iff for all x,y,zMhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetxyzM the following three identities hold:

    1. 1.

      (d)=0http://mathhub.info/smglom/mv/equal.omdoc?equal?equald iff x=yhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalxy (identity of indiscernibles),

    2. 2.

      (d)=(d)http://mathhub.info/smglom/mv/equal.omdoc?equal?equaldd (symmetry), and

    3. 3.

      (d)((d)+(d))http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethandhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additiondd (triangle inequality).

    We call M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd a metric space with base set MM and metric dd.


  • trivial divisor de

    1, 1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction, nn, and nhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn are known as the trivial divisors of nn. A divisor of nn that is not a trivial divisor is known as a proper or non-trivial divisor.


  • twelfth Smarandache constants

    Let (:f(NR)):fOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6ef be a function which satisfies the condition

    (ft)<(c((tατ((t!)))-(τ(((n-1)!)))))http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthanhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationfthttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionchttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationtαhttp://mathhub.info/smglom/smglom/numberofdivisorsfunction.omdoc?numberofdivisorsfunction?numberofdivisorsfunctionhttp://mathhub.info/smglom/numberfields/factorial.omdoc?factorial?factorialthttp://mathhub.info/smglom/smglom/numberofdivisorsfunction.omdoc?numberofdivisorsfunction?numberofdivisorsfunctionhttp://mathhub.info/smglom/numberfields/factorial.omdoc?factorial?factorialhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn

    for a natural numbers t>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethant, the given constants α>1http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanα, c>12http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanc and τ(n)http://mathhub.info/smglom/smglom/numberofdivisorsfunction.omdoc?numberofdivisorsfunction?numberofdivisorsfunctionn is the number of divisors of nn. The twelfth Smarandache constants are defind by:

    s12(f)http://mathhub.info/smglom/smglom/smarandacheconstant12.omdoc?smarandacheconstant12?twelfh-smarandache-constantf

    The sum converges.


  • two-sided prime

    A two-sided prime is both left-truncatable and right-truncatable.

    For example: 3137


  • undefined at de

    We call a partial function f:XYhttp://mathhub.info/smglom/sets/functions.omdoc?functions?partfunfXY undefined at xXhttp://mathhub.info/smglom/sets/set.omdoc?set?insetxX (write (f)=http://mathhub.info/smglom/mv/equal.omdoc?equal?equalfhttp://mathhub.info/smglom/sets/functions.omdoc?functions?undefd), iff (x,y)fhttp://mathhub.info/smglom/sets/set.omdoc?set?ninsethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairxyf for all yYhttp://mathhub.info/smglom/sets/set.omdoc?set?insetyY.


  • uniformly convergent

    Let AA be a set, B,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureBd a metric space, and (sequenceon[f]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonfn a sequence of functions (fi):ABhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funhttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselfiAB, then we call (sequenceon[f]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonfn uniformly convergent to f:ABhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfAB on AAhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efA, iff for every ϵ0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methanϵ, there exists a Nhttp://mathhub.info/smglom/sets/set.omdoc?set?insetNhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, such that for all xAhttp://mathhub.info/smglom/sets/set.omdoc?set?insetxOMFOREIGNscala.xml.Node$@6040b6ef and all nNhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methannN we have (d)<ϵhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthandϵ.

    Ist AA eine Menge, B,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureBd ein metrischer Raum und (sequenceon[f]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonfn eine Folge von Funktionen (fi):ABhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funhttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselfiAB, dann nennen wir (sequenceon[f]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonfn gleichmäïg konvergent gegen f:ABhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfAB auf AAhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efA, falls zu jedem ϵ0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methanϵ ein Nhttp://mathhub.info/smglom/sets/set.omdoc?set?insetNhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number existiert, so dass (d)<ϵhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthandϵ für alle xAhttp://mathhub.info/smglom/sets/set.omdoc?set?insetxOMFOREIGNscala.xml.Node$@6040b6ef und alle nNhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methannN.


  • unit fraction de

  • upper bound

    Let Shttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureS be a proset and TShttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetTS, then we call bShttp://mathhub.info/smglom/sets/set.omdoc?set?insetbS an upper bound of TT, iff (lessthan)http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthan for all tThttp://mathhub.info/smglom/sets/set.omdoc?set?insettT and an lower bound, iff (lessthan)http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthan for all tThttp://mathhub.info/smglom/sets/set.omdoc?set?insettT.


  • upper bound de

    We define the product over a sequence aiOMFOREIGNscala.xml.Node$@6040b6ef by

    (i=nmai):=(defined-piecewise(
    1wenn(n=m)
    )
    (
    (an)sonst
    )
    )
    http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdfromtonmiOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnmhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationOMFOREIGNscala.xml.Node$@6040b6ef

    The variable ii is called the index of multiplication and nn and mm the lower and upper bounds of the product respectively, together the specify the range of multiplication.

    There are variant product operators φaihttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdCollφOMFOREIGNscala.xml.Node$@6040b6ef and iSaihttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdInCollSiOMFOREIGNscala.xml.Node$@6040b6ef. The first one specifies the range of the product via a formula φφ in ii and the second one directly by giving a set SS.


  • upper bound de

    Summation is iterated addition, we define the sum over a sequnce aiOMFOREIGNscala.xml.Node$@6040b6ef by

    (i=nmai):=(defined-piecewise(
    0wenn(n=m)
    )
    (
    (an)sonst
    )
    )
    http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/numberfields/sum.omdoc?sum?sumnmiOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnmhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionOMFOREIGNscala.xml.Node$@6040b6ef

    The variable ii is called the index of summation and nn and mm the lower and upper bound of the sum respectively, together the specify the range of summation.

    There are variant summation operators (SumCollφ[i]ai)http://mathhub.info/smglom/numberfields/sum.omdoc?sum?SumCollφiOMFOREIGNscala.xml.Node$@6040b6ef and iSaihttp://mathhub.info/smglom/numberfields/sum.omdoc?sum?SumInCollSiOMFOREIGNscala.xml.Node$@6040b6ef. The first one specifies the range of the summation via a formula φφ in ii and the second one directly by giving a set SS.


  • upper limit Show Notations

    Ist M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd ein metrischer Raum und (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan eine Folge mit (ai)Mhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselaiM für ihttp://mathhub.info/smglom/sets/set.omdoc?set?insetihttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, so ist der

    • Limes superior limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (schreibe auch limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) definiert durch

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai
    • Limes inferior limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (schreibe auch limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) definiert durch

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai

    Let M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd be a metric space and (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan a sequence with (ai)Mhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselaiM for all ihttp://mathhub.info/smglom/sets/set.omdoc?set?insetihttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, then the

    • limit superior (also called supremum limit, superior limit, upper limit, or outer limit) limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (also written as limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) is defined as

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-superiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai
    • limit inferior (also called infimum limit, inferior limit, lower limit, or inner limit) limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai (also written as limi¯(ai)http://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai) is defined as

      (fundefeq[a]limi¯(ai))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqahttp://mathhub.info/smglom/calculus/limsupinf.omdoc?limsupinf?limit-inferiorihttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselai

  • vector de

    Let AA be a set, then the nn-dim Cartesian space Anhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?nCartSpaceAn over AA is (bsetst[an](a1,,an))http://mathhub.info/smglom/sets/set.omdoc?set?bsetstanhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlian. We call ((a1,,an))(An)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlianhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?nCartSpaceAn a vector.


  • vector product Show Notations de

    Let VV be an nn-dimensional vector space and {(v1,,vn)}http://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlivn be a basis of VV, we then define the vector product of the vectors w1,...,w(n-1)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?nseqliwhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn as

    (nfundefeqli[w]1)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?nfundefeqliw

  • vertex

    A pair of rays ABhttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?rayAB and AChttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?rayAC is called an angle with vertex AA. We write (AB),(AC)http://mathhub.info/smglom/sets/pair.omdoc?pair?pairhttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?rayABhttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?rayAC as ABChttp://mathhub.info/smglom/geometry/order-geometry.omdoc?order-geometry?angleABC.


  • vertices de

    A directed graph (also called digraph or oriented graph) is a pair V,Ehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE such that VV is a set and E(V×V)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetEhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsVV. We call VV the vertices (or nodes) and EE the edges of GG.


  • vertices de

    A graph is a pair V,Ehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE such that VV is a set and E(V×V)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetEhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsVV is a subset of the set of pairs from VV. We call VV the vertices (or nodes, points,junctions) and EE the edges (or lines, branches, arcs) of GG.


  • Wagstaff prime de

    A Wagstaff prime is a prime number pp of the form

    p=(((2q)+1)3)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationq

    where qq is another prime.


  • weakly prime de

    A prime number is called weakly prime if it becomes composite when any one of its digits is changed to every single other digit.


  • Wieferich prime de

    A Wieferich prime is a prime number pp such that p2http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationp divides (2(p-1))-1.0http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionp


  • Wilbraham-Gibbs constant

    The sine integral of πhttp://mathhub.info/smglom/numberfields/pinumber.omdoc?pinumber?pinumber is called the Wilbraham-Gibbs constant.

    (startswithSi(π)1851937)http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?startswithhttp://mathhub.info/smglom/smglom/sineintegralbig.omdoc?sineintegralbig?sine-integralhttp://mathhub.info/smglom/numberfields/pinumber.omdoc?pinumber?pinumber

  • Wilson number de

    A Wilson number is an integer nn such that (W(n))0modnhttp://mathhub.info/smglom/numberfields/congruence.omdoc?congruence?congruent-modulohttp://mathhub.info/smglom/smglom/wilsonquotient.omdoc?wilsonquotient?Wilson-Quotientnn where W(n)http://mathhub.info/smglom/smglom/wilsonquotient.omdoc?wilsonquotient?Wilson-Quotientn denotes the Wilson quotient (((n-1)!)+1)nhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/factorial.omdoc?factorial?factorialhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnn.


  • Wilson prime de

    A Wilson prime is a prime number pp such that p2http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationp divides ((p-1)!)+1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/factorial.omdoc?factorial?factorialhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionp.


  • Wilson quotient de

    For an integer mm the Wilson quotient W(m)http://mathhub.info/smglom/smglom/wilsonquotient.omdoc?wilsonquotient?Wilson-Quotientm is defind by

    (W(m))=((((m-1)!)+1)m)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/smglom/wilsonquotient.omdoc?wilsonquotient?Wilson-Quotientmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/factorial.omdoc?factorial?factorialhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionmm

  • Wolstenholme prime de

    A prime pp is called a Wolstenholme prime iff the following condition holds:

    (𝒞(p-1)((2p)-1))1mod(p4)http://mathhub.info/smglom/numberfields/congruence.omdoc?congruence?congruent-modulohttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficienthttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationp

  • Woodall number Show Notations de

  • zero de

    A structure Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR is called a ring, if Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR is a Abelian group (called the additive structure), Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR is a monoid (called the multiplicative structure), and Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR is a ringoid.

    We call 0 the zero of the ring and 1 the one of the ring.


  • zero de

    Let KK be a field, 0Khttp://mathhub.info/smglom/sets/set.omdoc?set?insetK its additive unit, and f:DKhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfDK for a set DD, then we call any z(f-1(0))http://mathhub.info/smglom/sets/set.omdoc?set?insetzhttp://mathhub.info/smglom/sets/image.omdoc?image?pre-imagef a zero of ff.


  • zerofree de

    An integer whose decimal digits contain no zeros is said to be zerofree.


  • alt Show Notations ro de en

    AA kümesi, BB kümesinin alt kümesidir (ABhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?proper-subsetAB olarak yazılır), eğer her bir xAhttp://mathhub.info/smglom/sets/set.omdoc?set?insetxA BB’nin de elemanı ise.


  • ayrık de ro en

    İki küme AA ve BB’ye, ayrık denir, eğer (AB)=http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/sets/intersection.omdoc?intersection?intersectionABhttp://mathhub.info/smglom/sets/emptyset.omdoc?emptyset?eset ise.

    Bir mümeler ailesi çift olarak ayrık ya da karşılıklı olarak ayrık dır, eğer herhangi iki tanesi ayrık ise.


  • birleşim Show Notations de ro ru

    II bir küme ve (bsetst[i]Si)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstiOMFOREIGNscala.xml.Node$@6040b6ef bir kümeler ailesi olsun. Bu durumda, SS topluluğu üzerindeki birleşim iISihttp://mathhub.info/smglom/sets/union.omdoc?union?munionCollectionIiOMFOREIGNscala.xml.Node$@6040b6ef şudur: (bsetst[x]x)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstxx için.


  • birleşim Show Notations de ro ru

    AA ve BB iki küme olsun. Bu durumda, AA ve BB’nin birleşimi ABhttp://mathhub.info/smglom/sets/union.omdoc?union?unionAB, (bsetst[x]x)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstxx olarak tanımlanır.


  • Boş küme Show Notations en de ro

    Boş küme http://mathhub.info/smglom/sets/emptyset.omdoc?emptyset?eset (http://mathhub.info/smglom/sets/emptyset.omdoc?emptyset?eset olarak da yazılır) hiç elemanı olmayan k umedir.


  • düzgün alt küme Show Notations ro en

    AA kümesi, BB kümesinin bir düzgün alt kümesidir (ABhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?supersetAB olarak yazılır), eğer BAhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetBA ise.


  • düzgün alt küme Show Notations ro de en

    AA kümesi, BB kümesinin düzgün alt kümesidir (ABhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetAB olarak yazılır), eğer ABhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?proper-subsetAB ise ama ABhttp://mathhub.info/smglom/sets/set.omdoc?set?nsetequalAB ise.


  • eşit Show Notations ro de ru

    İki küme AA ve BB eşittir (ABhttp://mathhub.info/smglom/sets/set.omdoc?set?setequalAB olarak yazılır), eğer her ikisi de aynı elemanlara sahip iseler.


  • fark kümesi Show Notations de ro bg

    AA ve BB iki küme olsun. Bu durumda, AA ve BB’nin fark kümesi A\Bhttp://mathhub.info/smglom/sets/setdiff.omdoc?setdiff?set-differenceAB (bsetst[x]x)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstxx dir.


  • karşılıklı olarak ayrık ro de en

    Bir mümeler ailesi çift olarak ayrık ya da karşılıklı olarak ayrık dır, eğer herhangi iki tanesi ayrık ise.


  • kesişim Show Notations de ro en

    II bir küme ve SiOMFOREIGNscala.xml.Node$@6040b6ef II tarafından indekslenmiş bir küme ailesi olsun. Bu durumda II üzerine kesişim (bsetst[x]x)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstxx iISihttp://mathhub.info/smglom/sets/intersection.omdoc?intersection?mintersectCollectionIiOMFOREIGNscala.xml.Node$@6040b6ef dir.


  • kesişim Show Notations de ro en

    AA ve BB iki küme olsun. Bu durumda, AA ve BB’nin kesişim i ABhttp://mathhub.info/smglom/sets/intersection.omdoc?intersection?intersectionAB, (bsetst[x]x)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstxx dir.


  • çift Show Notations en ro de

    AA ve BB iki küme olsun. Bu durumda, AA ve BB çifti A×Bhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAB, (bsetst[ab]a,b)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstabhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairab olarak tanımlanır, (a,b)(A×B)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairabhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAB’ya bir çift denir.


  • çift Show Notations en ro de

    AA ve BB iki küme olsun. Bu durumda, AA ve BB çifti A×Bhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAB, (bsetst[ab]a,b)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstabhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairab olarak tanımlanır, (a,b)(A×B)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairabhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAB’ya bir çift denir.


  • çift olarak ayrık ro de en

    Bir mümeler ailesi çift olarak ayrık ya da karşılıklı olarak ayrık dır, eğer herhangi iki tanesi ayrık ise.


  • üssü küme Show Notations de ro bg

    AA bir küme olsun. Bu durumda, AA’nin üssü kümesi 𝒫(A)http://mathhub.info/smglom/sets/powerset.omdoc?powerset?powersetA, (bsetst[S]S)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstSS dir.


  • üst küme Show Notations ro de en

    AA kümesi, BB kümesinin bir üst kümesidir (ABhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?proper-supersetAB olarak yazılır), eğer BAhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?proper-subsetBA ise.


  • mal differeinzierbar en

    Wir nennen ff nn mal differeinzierbar in aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM, falls dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa als Grenzwert existiert. Analog nennen wir ff nn mal differenzierbar auf MM, falls dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx existiert und auf MM total ist.

    Wir nennen ff unendlich differenzierbar oder glatt auf aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM oder MM, wenn dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa beziehungsweise dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx für alle nhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number existieren.


  • mal differenzierbar en

    Wir nennen ff nn mal differeinzierbar in aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM, falls dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa als Grenzwert existiert. Analog nennen wir ff nn mal differenzierbar auf MM, falls dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx existiert und auf MM total ist.

    Wir nennen ff unendlich differenzierbar oder glatt auf aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM oder MM, wenn dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa beziehungsweise dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx für alle nhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number existieren.


  • -Abschluss en

    Ist pp eine Eigenschaft und R(A×B)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetRhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAB eine Relation, dann nennen wir die kleinste (bezüglich subsethttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subset) Relation RRhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?supersetOMFOREIGNscala.xml.Node$@6040b6efR die Eigenschaft pp hat den pp-Abschluss von RR.


  • -additiv

    Eine Folge von Zahlen ist ss-additiv, wenn jede Zahl der Folge, nach den 2shttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplications Anfangsgliedern, auf genau ss Arten als Summe zweier vorheriger Zahlen der Folge dargestellt werden kann.


  • -dimenionale Cartesische Raum Show Notations en

    Der nn-dimenionale Cartesische Raum Anhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?nCartSpaceAn über einer Menge AA ist definiert als (bsetst[an](a1,,an))http://mathhub.info/smglom/sets/set.omdoc?set?bsetstanhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlian. Wir nennen ein element ((a1,,an))(An)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlianhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?nCartSpaceAn einen Vektor.


  • -fache Cartesische Product Show Notations en

    Sei AA eine Familie von Mengen dann definieren wir das nn-fache Cartesische Product A1×...×Anhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?ncartliAn als (bsetst[an](a1,,an))http://mathhub.info/smglom/sets/set.omdoc?set?bsetstanhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlian, und nennen ((a1,,an))(A1×...×An)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlianhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?ncartliAn nn-Tupel.

    Wir nennen die Funktion (projectioni):(A1×...×An)Ai;((a1,,an))aihttp://mathhub.info/smglom/sets/functions.omdoc?functions?funsuchthathttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?projectionihttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?ncartliAnOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlianOMFOREIGNscala.xml.Node$@6040b6ef die iite Projektion.


  • -gleich en

    Wir nennen eine Formel AA eine alphabetische Variante von BB (oder http://mathhub.info/smglom/smglom/alpharenaming.omdoc?alpharenaming?alphaeqRel-gleich; schreibe (alphaeqA)Bhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/alpharenaming.omdoc?alpharenaming?alphaeqAB), wenn BB aus AA hervorgeht durch systematische Umbenennung gebundener Variablen.


  • -perfekt en

  • -potente Zahl en

    Eine kk-potente Zahl ist eine natürliche Zahl mm, so dass für jeden Primteiler pp von mm auch pkhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationpk Teiler von mm ist.


  • -potenzglatt en

    Eine natürliche Zahl heißt SS-potenzglatt bezüglich einer Schranke SS, wenn in ihrer Primfaktorzerlegung nur Primpotenzen kleiner oder gleich SS vorkommen. Das heißt, für jeden Primfaktor pp, der kk mal vorkommt, gilt: (pk)<Shttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthanhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationpkS.


  • -regulär en

    Wenn in einem Graph GG alle Ecken denselben Grad aufweisen, sagen wir kk, dann nennt man GG kk-regulär, oder schlicht regulär.

    Ein 3-regulärer Graph wird auch kubischer Graph genannt.


  • -Simplex en

    Ein kk-Simplex ist ein kk-dimensionales Polytop das als die konvexe Hülle von k+1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionk affin unabhängigen Punkten in khttp://mathhub.info/smglom/linear-algebra/nspace.omdoc?nspace?Euclidean-spacek gegeben ist.


  • -Tupel Show Notations en

    Sei AA eine Familie von Mengen dann definieren wir das nn-fache Cartesische Product A1×...×Anhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?ncartliAn als (bsetst[an](a1,,an))http://mathhub.info/smglom/sets/set.omdoc?set?bsetstanhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlian, und nennen ((a1,,an))(A1×...×An)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlianhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?ncartliAn nn-Tupel.

    Wir nennen die Funktion (projectioni):(A1×...×An)Ai;((a1,,an))aihttp://mathhub.info/smglom/sets/functions.omdoc?functions?funsuchthathttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?projectionihttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?ncartliAnOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlianOMFOREIGNscala.xml.Node$@6040b6ef die iite Projektion.


  • -Ulam-Folge en

    Die u,vhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairuv-Ulam-Zahlen Unhttp://mathhub.info/smglom/numbers/uvulamnumber.omdoc?uvulamnumber?uvulamnumbern bilden eine ganzzahlige Folge. Die u,vhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairuv-Ulam-Folge beginnt mit (U1)=uhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/uvulamnumber.omdoc?uvulamnumber?uvulamnumberu und (U2)=vhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/uvulamnumber.omdoc?uvulamnumber?uvulamnumberv. Für n>2http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethann wird dann Unhttp://mathhub.info/smglom/numbers/uvulamnumber.omdoc?uvulamnumber?uvulamnumbern definiert als die kleinste ganze Zahl, die sich auf genau eine Weise als Summe zweier verschiedener vorhergehender Ulam-Zahlen darstellen lässt.


  • -Ulam-Zahlen en

    Die u,vhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairuv-Ulam-Zahlen Unhttp://mathhub.info/smglom/numbers/uvulamnumber.omdoc?uvulamnumber?uvulamnumbern bilden eine ganzzahlige Folge. Die u,vhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairuv-Ulam-Folge beginnt mit (U1)=uhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/uvulamnumber.omdoc?uvulamnumber?uvulamnumberu und (U2)=vhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/uvulamnumber.omdoc?uvulamnumber?uvulamnumberv. Für n>2http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethann wird dann Unhttp://mathhub.info/smglom/numbers/uvulamnumber.omdoc?uvulamnumber?uvulamnumbern definiert als die kleinste ganze Zahl, die sich auf genau eine Weise als Summe zweier verschiedener vorhergehender Ulam-Zahlen darstellen lässt.


  • abgeschlossen o en

    Ein topologischer Raum X,Ohttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureXOMFOREIGNscala.xml.Node$@6040b6ef ist eine Menge XX zusammen mit einer Familie O(𝒫(X))http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/sets/powerset.omdoc?powerset?powersetX, so daß

    1. 1.

      ,XOhttp://mathhub.info/smglom/sets/set.omdoc?set?minsethttp://mathhub.info/smglom/sets/emptyset.omdoc?emptyset?esetXOMFOREIGNscala.xml.Node$@6040b6ef.

    2. 2.

      (SSs)Ohttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/union.omdoc?union?munionCollectionOMFOREIGNscala.xml.Node$@6040b6efSsOMFOREIGNscala.xml.Node$@6040b6ef falls SShttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetSOMFOREIGNscala.xml.Node$@6040b6ef.

    3. 3.

      (SSs)Ohttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/intersection.omdoc?intersection?mintersectCollectionOMFOREIGNscala.xml.Node$@6040b6efSsOMFOREIGNscala.xml.Node$@6040b6ef falls SShttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetSOMFOREIGNscala.xml.Node$@6040b6ef endlich.

    Dann heißt OOMFOREIGNscala.xml.Node$@6040b6ef eine Topologie) auf XX. Elemente der Topologie OOMFOREIGNscala.xml.Node$@6040b6ef heißen offene Mengen und ihre Komplemente abgeschlossen oder einfach geschlossen . Eine Teilmenge von XX kann weder gesclossen noch offen, oder gesclossen, oder offen oder beides.


  • Ableitung Show Notations en

    Ist f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN differenzierbar auf MM, so definieren wir die Ableitungsfunktion (Dxf):MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfxMN von ff nach xx als

    (fundefeq[fxa]Dxf(a))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfxahttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa

    Die Abhängigkeit von xx ist bleibt dieser Notation (Lagrange Notation) implizit. In Leibniz Notation schreiben wir die Ableitungsfunktion von ff nach xx als Dxfhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfx. In Eulers Notation schreiben wir Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa und in Newtonnotation Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa.

    Wir nennen f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN differenzierbar an einem Häufungspunkt pMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetpM, falls der Grenzwert

    d:=(limxp((dB)(dA)))http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationdhttp://mathhub.info/smglom/calculus/functionlimit.omdoc?functionlimit?limitpxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6ef

    existiert. Dieser Grenzwert heißt Ableitung von ff nach xx an der Stelle pp. Wir nennen ff differenzierbar auf MM, falls ff differenzierbar an jedem mMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetmM ist.


  • Ableitungsfunktion en

    Ist f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN differenzierbar auf MM, so definieren wir die Ableitungsfunktion (Dxf):MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfxMN von ff nach xx als

    (fundefeq[fxa]Dxf(a))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfxahttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa

    Die Abhängigkeit von xx ist bleibt dieser Notation (Lagrange Notation) implizit. In Leibniz Notation schreiben wir die Ableitungsfunktion von ff nach xx als Dxfhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfx. In Eulers Notation schreiben wir Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa und in Newtonnotation Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa.


  • Abrundung Show Notations en

    Die Abrundung (oder der ganze Teil)

    r
    http://mathhub.info/smglom/numberfields/ceilingfloor.omdoc?ceilingfloor?floorr einer reellen Zahl rr ist die größte ganze Zahl, die nicht größer ist als rr. Die Aufrundungsfunktion wird auch die Gaussklammer genannt; dann wird sie als r
    http://mathhub.info/smglom/numberfields/ceilingfloor.omdoc?ceilingfloor?floorr geschrieben.


  • absolut nicht-normal en

    Eine reelle Zahl ist absolut nicht-normal oder absolut unnormal, wenn sie in keiner Basis einfach normal ist.


  • absolut unnormal en

    Eine reelle Zahl ist absolut nicht-normal oder absolut unnormal, wenn sie in keiner Basis einfach normal ist.


  • Absolutbetrag Show Notations en

    Der Absolutbetrag |r|http://mathhub.info/smglom/numberfields/absolutevalue.omdoc?absolutevalue?absolute-valuer einer reellen Zahl rr ist definiert als (defined-piecewise(
    rwenn(r0)
    )
    (
    (r)sonst
    )
    )
    http://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecerhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methanrhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionr
    .


  • Abstandsfunktion en

    Für eine Mente MM nennen wir eine Funktion d:(M×M)http://mathhub.info/smglom/sets/functions.omdoc?functions?fundhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsMMhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number eine Abstandsfunktion (or Metrik ) auf MM, falls die folgenden drei Eigenschaften für alle x,y,zMhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetxyzM gelten:

    1. 1.

      (d)=0http://mathhub.info/smglom/mv/equal.omdoc?equal?equald gdw. x=yhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalxy (Definitheit),

    2. 2.

      (d)=(d)http://mathhub.info/smglom/mv/equal.omdoc?equal?equaldd (Symmetrie) und

    3. 3.

      (d)((d)+(d))http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethandhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additiondd (Dreiecksungleichung).

    Wir nennen M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd einen metrischen Raum mit Grundmenge MM und Metrik dd.


  • Abundancy en

    Die Abundancy einer ganzen Zahl nn ist das Verhältnis (σ1(n))nhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/smglom/sumofdivisorsfunction.omdoc?sumofdivisorsfunction?sumdiv-functionnn. Dabei ist σ1(n)http://mathhub.info/smglom/smglom/sumofdivisorsfunction.omdoc?sumofdivisorsfunction?sumdiv-functionn die Teilersummenfunktion.


  • Abundanz en

    Die Abundanz einer positiven ganzen Zahl nn ist der Wert (σ1(n))-(2n)http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/smglom/sumofdivisorsfunction.omdoc?sumofdivisorsfunction?sumdiv-functionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationn.


  • abzählbar en

    Wir nennen eine Menge AA abzählbar, wenn es eine surjektive Function f:Ahttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfAhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number gibt.


  • abzählbar unendlich en

    Wir nennen eine Menge AA abzählbar unendlich, wenn es eine bijektive Function f:Ahttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfAhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number gibt.


  • Achilles Zahl en

    Eine Achilles Zahl ist eine potente Zahl aber keine perfekte Potenz.


  • achte Smarandache-Konstante Show Notations en

    Die achte Smarandache-Konstante http://mathhub.info/smglom/smglom/smarandacheconstant8.omdoc?smarandacheconstant8?eighth-smarandache-constant ist für eine natürliche Zahl kk definiert als

    http://mathhub.info/smglom/smglom/smarandacheconstant8.omdoc?smarandacheconstant8?eighth-smarandache-constant

    Dabei ist S(n)http://mathhub.info/smglom/smglom/smarandachefunction.omdoc?smarandachefunction?smarandache-functionn die Smarandache-Funktion.

    Die Summe konvergiert.


  • Addition Show Notations

    Die arithmetischen Operationen sind Addition additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?addition, Subtraktion, subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction, Multiplikation multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplication, Division divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?division und Exponentiation.


  • additive Struktur en

    Eine Struktur Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR heißt Ring, wenn Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR eine Abelsch e Gruppe ist (die additive Struktur), Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR ein Monoid (die multiplikative Struktur) ist, sowie Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR is a Ringoid.

    Wir nennen 0 die Null des Rings und entsprechend 1 die Eins.


  • adjazent en

    Zwei Ecken xx und yy in einem Graph GG sind adjazent, oder Nachbarn, wenn x,yhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairxy eine Kante von GG ist.

    Zwei Kanten efhttp://mathhub.info/smglom/mv/equal.omdoc?equal?notequalef sind adjazent, wenn sie ein Ende gemeinsam haben.


  • affin unabhängig en

    Wir nennen eine Menge (bsetst[i]ui)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstiOMFOREIGNscala.xml.Node$@6040b6ef von Punkten in nhttp://mathhub.info/smglom/linear-algebra/nspace.omdoc?nspace?Euclidean-spacen affin unabhängig, wenn die Menge (bsetst[i]u0-ui)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstihttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6ef linear unabhängig sind.


  • Aliquot-Folge en

    Eine Aliquot-Folge startet mit einer positiven ganzen Zahl kk. Jedes weitere Glied ist die Summe der echtenTeiler des vorhergehenden Gliedes.


  • all-Harshad-Zahl en

    Eine all-Harshad-Zahl oder eine all-Niven-Zahl ist eine ganze Zahl, die in jeder beliebigen Zahlenbasis eine Harshad-Zahl ist. Es gibt nur vier all-Harshad-Zahlen: 1, 2, 4, und 6.


  • all-Niven-Zahl en

    Eine all-Harshad-Zahl oder eine all-Niven-Zahl ist eine ganze Zahl, die in jeder beliebigen Zahlenbasis eine Harshad-Zahl ist. Es gibt nur vier all-Harshad-Zahlen: 1, 2, 4, und 6.


  • alphabetische Variante en

    Wir nennen eine Formel AA eine alphabetische Variante von BB (oder http://mathhub.info/smglom/smglom/alpharenaming.omdoc?alpharenaming?alphaeqRel-gleich; schreibe (alphaeqA)Bhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/alpharenaming.omdoc?alpharenaming?alphaeqAB), wenn BB aus AA hervorgeht durch systematische Umbenennung gebundener Variablen.


  • alternierende harmonische Serie en

    Die alternierende harmonische Serie ist die Serie

    (infinite-sum1[n](multi-relation-expression((1)(n+1))nequal((((1-(12))+(13))-(14))+(15))-equalln(2)))http://mathhub.info/smglom/calculus/infinitesum.omdoc?infinitesum?infinite-sumnhttp://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnnhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/calculus/naturallogarithm.omdoc?naturallogarithm?natural-logarithm

  • annährend gleich Show Notations en

    Wir nennen zwei mathematische Objekte aa and bb annährend gleich, (schreibe abhttp://mathhub.info/smglom/mv/approxeq.omdoc?approxeq?approximately-equalab), wenn die einzigen Eigenschaften, die sie auseinanderhalten, weniger relevant sind in der aktuellen Situation.


  • antireflexiv en

    Eine Relation R(A×A)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetRhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAA heißt

    • reflexiv auf AA, wenn (a,a)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairaaR für alle aAhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaA, und

    • irreflexiv (or antireflexiv) auf AA, wenn (a,a)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?ninsethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairaaR für alle aAhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaA.


  • antisymmetrisch en

    Eine Relation R(A×A)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetRhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAA heißt

    • symmetrisch auf AA, falls (b,a)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairbaR für alle a,bAhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabA mit (a,b)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairabR.

    • asymmetrisch auf AA, falls (b,a)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?ninsethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairbaR für alle a,bAhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabA mit (a,b)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairabR.

    • antisymmetrisch auf AA, falls a=bhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalab wenn (a,b)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairabR und (b,a)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairbaR.


  • Argumentbereich Show Notations en

    Eine Relation f(X×Y)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetfhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsXY, heißt partielle Funktion mit Argumentbereich XX (schreibe 𝐝𝐨𝐦(f)http://mathhub.info/smglom/sets/functions.omdoc?functions?domainf) und Wertebereich YY (schreibe 𝐜𝐨𝐝𝐨𝐦(f)http://mathhub.info/smglom/sets/functions.omdoc?functions?codomainf), wenn es für jedes xXhttp://mathhub.info/smglom/sets/set.omdoc?set?insetxX höchstens ein yYhttp://mathhub.info/smglom/sets/set.omdoc?set?insetyY gibt mit (x,y)fhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairxyf.

    Wir schreiben f:XY;xyhttp://mathhub.info/smglom/sets/functions.omdoc?functions?partfunsuchthatfXYxy und (f)=yhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalfy wenn (x,y)fhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairxyf.


  • arithmetische Mittel Show Notations en

    Das arithmetische Mittel einer Menge M=({(a1,,an)})http://mathhub.info/smglom/mv/equal.omdoc?equal?equalMhttp://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlian ist definiert als AMhttp://mathhub.info/smglom/numberfields/arithmeticmean.omdoc?arithmeticmean?arithmetic-meanM.


  • arithmetischen Operationen

    Die arithmetischen Operationen sind Addition additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?addition, Subtraktion, subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction, Multiplikation multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplication, Division divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?division und Exponentiation.


  • arithmetischen Operationen

    Die arithmetischen Operationen sind Addition additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?addition, Subtraktion, subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction, Multiplikation multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplication, Division divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?division und Exponentiation.


  • asymmetrisch en

    Eine Relation R(A×A)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetRhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAA heißt

    • symmetrisch auf AA, falls (b,a)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairbaR für alle a,bAhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabA mit (a,b)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairabR.

    • asymmetrisch auf AA, falls (b,a)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?ninsethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairbaR für alle a,bAhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabA mit (a,b)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairabR.

    • antisymmetrisch auf AA, falls a=bhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalab wenn (a,b)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairabR und (b,a)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairbaR.


  • asymptotische Dichte Show Notations en

    Eine Teilmenge AA der positiven ganzen Zahlen hat eine asymptotische Dichte (oder natürliche Dichte) d(A)http://mathhub.info/smglom/smglom/asymptoticdensity.omdoc?asymptoticdensity?asymptotic-densityA, wobei (multi-relation-expression0lessthand(A)lessthan1)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthanhttp://mathhub.info/smglom/smglom/asymptoticdensity.omdoc?asymptoticdensity?asymptotic-densityAhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthan gilt, wenn der Grenzwert existiert

    d(A)http://mathhub.info/smglom/smglom/asymptoticdensity.omdoc?asymptoticdensity?asymptotic-densityA

    anhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationan ist die Anzahl der Elemente von AA, die nicht größer als nn sind.


  • Aufrundung Show Notations en

    Die Abrundung (oder der ganze Teil)

    r
    http://mathhub.info/smglom/numberfields/ceilingfloor.omdoc?ceilingfloor?floorr einer reellen Zahl rr ist die größte ganze Zahl, die nicht größer ist als rr. Die Aufrundungsfunktion wird auch die Gaussklammer genannt; dann wird sie als r
    http://mathhub.info/smglom/numberfields/ceilingfloor.omdoc?ceilingfloor?floorr geschrieben.

    Die Aufrundung ]]r[[http://mathhub.info/smglom/numberfields/ceilingfloor.omdoc?ceilingfloor?ceilingr einer reellen Zahl rr ist die kleinste ganze Zahl, die nicht kleiner ist als rr.


  • azyklisch

    Ist G=(V,E)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalGhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE ein gerichteter Graph, so nennen wir

    • einen Pfad pp in GG zyklisch (auch einen Zykel oder eine Schleife), falls (start(p))=(end(p))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pstartphttp://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pendp.

    • einen Zykel (v0,,vn)http://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlivn einfach, wenn vivjhttp://mathhub.info/smglom/mv/equal.omdoc?equal?notequalOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6ef für alle i,j1nhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?mbetweeneeijn mit ijhttp://mathhub.info/smglom/mv/equal.omdoc?equal?notequalij.

    • GG azyklisch, wenn GG keinen Zykel enthält.


  • Bailey-Borwein-Plouffe-Formel en

    Die Bailey-Borwein-Plouffe-Formel (BBP-Formel) ist eine Reihe zur Berechnung von πhttp://mathhub.info/smglom/numberfields/pinumber.omdoc?pinumber?pinumber:

    πhttp://mathhub.info/smglom/numberfields/pinumber.omdoc?pinumber?pinumber

  • Balance-Primzahl en

    Eine Balance-Primzahl ist eine Primzahl, die gleich dem arithmetischen Mittel der nächstkleineren und der nächstgrößeren Primzahl ist.

    (pn)=(((p(n-1))+(p(n+1)))2)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?prime-numbernhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?prime-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?prime-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionn

    Dabei ist pnhttp://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?prime-numbern die nn-te Primzahl.


  • Barriere en

    Eine reelle Zahl nn wird Barriere Qeiner zahlentheoretischen Funktion fmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationfm genant, wenn (m+(fm))<nhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthanhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationfmn für alle m>nhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanmn.


  • Baum en

  • Baxter-Hickerson-Funktion

    Die Baxter-Hickerson-Funktion ist für nicht-negative ganze Zahlen nn definiert als

    (fn)=((13)(((210(5n))-(10(4n)))+(210(3n))+(10(2n))+(10n)+1))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationfnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn

    Sie erzeugt Zahlen, deren Kuben nicht die Ziffer 0 enthalten.


  • BBP-Formel en

    Die Bailey-Borwein-Plouffe-Formel (BBP-Formel) ist eine Reihe zur Berechnung von πhttp://mathhub.info/smglom/numberfields/pinumber.omdoc?pinumber?pinumber:

    πhttp://mathhub.info/smglom/numberfields/pinumber.omdoc?pinumber?pinumber

  • Befreundete Zahlen en

    Befreundete Zahlen sind ein Paar natürlicher Zahlen, deren Summen ihrer Teiler (außer den Zahlen selbst) jeweils die andere Zahl ergibt.


  • Beharrlichkeit en

    Die Anzahl der notwendigen Schritte, um durch wiederholte Querprodukte zu einer einstelligen Zahl zu gelangen, nennt man Beharrlichkeit der Zahl.


  • Beidseitig trunkierbare Primzahlen

    Beidseitig trunkierbare Primzahlen sind sowohl links- als auch rechtstrunkierbare Primzahlen.

    Zum Beispiel: 3137


  • Beschränkung Show Notations en

    Ist f:ABhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfAB eine Funktion und CAhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetCA, so nennen wir (fundefeq[fC]f|C)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfChttp://mathhub.info/smglom/sets/function-restriction.omdoc?function-restriction?restrictionfC die Beschränkung von ff auf CC.


  • bijektiv en

    Eine Funktion f:SThttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfST heißt bijektiv, wenn ff injektiv and surjektiv ist.


  • Bild Show Notations en

    Sei f:ABhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfAB eine Funktion, AAhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efA und BBhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efB, dann nennen wir

    • (fundefeq[fname.cvar.2]f(A))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfname.cvar.2http://mathhub.info/smglom/sets/image.omdoc?image?imageoffOMFOREIGNscala.xml.Node$@6040b6ef das Bild von AOMFOREIGNscala.xml.Node$@6040b6ef unter ff,

    • (fundefeq[f]𝐈𝐦(f))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfhttp://mathhub.info/smglom/sets/image.omdoc?image?imagef das Bild von ff, und

    • (fundefeq[fname.cvar.6]f-1(B))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfname.cvar.6http://mathhub.info/smglom/sets/image.omdoc?image?pre-imagefOMFOREIGNscala.xml.Node$@6040b6ef das Urbild von BOMFOREIGNscala.xml.Node$@6040b6ef unter ff.


  • Bild Show Notations en

    Sei f:ABhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfAB eine Funktion, AAhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efA und BBhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efB, dann nennen wir

    • (fundefeq[fname.cvar.2]f(A))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfname.cvar.2http://mathhub.info/smglom/sets/image.omdoc?image?imageoffOMFOREIGNscala.xml.Node$@6040b6ef das Bild von AOMFOREIGNscala.xml.Node$@6040b6ef unter ff,

    • (fundefeq[f]𝐈𝐦(f))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfhttp://mathhub.info/smglom/sets/image.omdoc?image?imagef das Bild von ff, und

    • (fundefeq[fname.cvar.6]f-1(B))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfname.cvar.6http://mathhub.info/smglom/sets/image.omdoc?image?pre-imagefOMFOREIGNscala.xml.Node$@6040b6ef das Urbild von BOMFOREIGNscala.xml.Node$@6040b6ef unter ff.


  • Binomialkoeffizient Show Notations en

    Der Binomialkoeffizient 𝒞knhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficientnk ist definiert als die Anzahl der kk-elementigen Teilmengen einer nn-elementigen Menge.


  • binäre Logarithmus en

    Der binäre Logarithmus is der Logarithmus zur Basis 2.


  • Bögen en

    Ein gerichteter Graph (auch Digraph) ist ein Paar V,Ehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE aus einer Menge VV und einer Menge E(V,V)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetEhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairVV geordneter Paare über VV. Wir nennen VV die Knoten und EE die Kanten (auch Bögen) von GG.


  • Cabtaxi-Zahl Show Notations en

    Die nn-te Cabtaxi-Zahl Cabtaxi(n)http://mathhub.info/smglom/numbers/cabtaxinumber.omdoc?cabtaxinumber?cabtaxi-numbern ist definiert als die kleinste positive ganze Zahl, die auf nn verschiedene Arten als Summe oder Differenz zweier Kubikzahlen (einschließlich 0) dargestellt werden kann.

    Zum Beispiel:

    (multi-relation-expressionCabtaxi(2)equal91equal(33)+(43)equal(63)+(53))http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numbers/cabtaxinumber.omdoc?cabtaxinumber?cabtaxi-numberhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiation

  • Cahen-Konstante

    Sei snhttp://mathhub.info/smglom/numbers/sylvestersequence.omdoc?sylvestersequence?sylvestersequencen die Sylvester-Folge.

    Die Cahen-Konstante Chttp://mathhub.info/smglom/smglom/cahenconstantegyptian.omdoc?cahenconstantegyptian?cahenconstegyptian ist ein ägyptischerBruch, gebildet aus einer unendlichen Reihe von Stammbrüchen, deren Nenner die geradzahligen Elemente der Sylvester-Folge sind:

    Chttp://mathhub.info/smglom/smglom/cahenconstantegyptian.omdoc?cahenconstantegyptian?cahenconstegyptian

  • Cahen-Konstante i Show Notations en

    Sei snhttp://mathhub.info/smglom/numbers/sylvestersequence.omdoc?sylvestersequence?sylvestersequencen die Sylvester-Folge.

    Die Cahen-Konstante ist eine unendliche Reihe von Stammbrüchen mit alternierenden Vorzeichen

    𝐶:=(infinite-sum0[i]((1)i)((si)-1))http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/smglom/cahenconstant.omdoc?cahenconstant?cahenconstanthttp://mathhub.info/smglom/calculus/infinitesum.omdoc?infinitesum?infinite-sumihttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionihttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numbers/sylvestersequence.omdoc?sylvestersequence?sylvestersequencei

  • Carmichael-Zahl

    Eine zusammengesetzte natürliche Zahl nn heißt Carmichael-Zahl, falls für alle zu nn teilerfremden Zahlen aa die folgende Kongruenz erfüllt ist:

    (a(n-1))1modnhttp://mathhub.info/smglom/numberfields/congruence.omdoc?congruence?congruent-modulohttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationahttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnn

  • Carol-Zahl en

    Eine Carol-Zahl ist eine ganze Zahl der Form

    ((4n)-(2(n+1))-1)=((((2n)-1)2)-2)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn

    mit der positiven ganzen Zahl nn.


  • Cassini-Identität en

    Die Cassini-Identität ist eine Identität für ganze Zahlen n>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethann und die Fibonacci-Zahlen.

    (((F(n-1))(F(n+1)))-((Fn)2))=((1)n)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numbers/fibonaccinumbers.omdoc?fibonaccinumbers?Fibonacci-numbershttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numbers/fibonaccinumbers.omdoc?fibonaccinumbers?Fibonacci-numbershttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numbers/fibonaccinumbers.omdoc?fibonaccinumbers?Fibonacci-numbersnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn

  • Catalan-Identität en

    Die Catalan-Identität ist eine Identität für ganze Zahlen (multi-relation-expressionnmorethanrmorethan0)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionnhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanrhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethan, und die Fibonacci-Zahlen.

    (((Fn)2)-((F(n-r))(F(n+r))))=(((1)(n-r))((Fr)2))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numbers/fibonaccinumbers.omdoc?fibonaccinumbers?Fibonacci-numbersnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numbers/fibonaccinumbers.omdoc?fibonaccinumbers?Fibonacci-numbershttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnrhttp://mathhub.info/smglom/numbers/fibonaccinumbers.omdoc?fibonaccinumbers?Fibonacci-numbershttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnrhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnrhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numbers/fibonaccinumbers.omdoc?fibonaccinumbers?Fibonacci-numbersr

  • Catalan-Mersenne-Zahlen en

    Die Catalan-Mersenne-Zahlen sind Mersenne-Zahlen, der Form (cn):=(2(c(n-1)))http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/numbers/catalanmersennenumber.omdoc?catalanmersennenumber?catalanmersennenumbernhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numbers/catalanmersennenumber.omdoc?catalanmersennenumber?catalanmersennenumberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn.


  • Catalan-Zahlen en

    Die Catalan-Zahlen bilden eine Folge von natürlichen Zahlen, die wie folgt definiert ist

    (multi-relation-expressionCnequal(𝒞n(2n))-(𝒞(n+1)(2n))equal(1(n+1))(𝒞n(2n))equal((2n)!)((n+1)!))http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numbers/catalannumber.omdoc?catalannumber?catalannumbernhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficienthttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnnhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficienthttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficienthttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnnhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/factorial.omdoc?factorial?factorialhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnhttp://mathhub.info/smglom/numberfields/factorial.omdoc?factorial?factorialhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionn

    für n0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methann.


  • Cauchyfolge en

    In einem metrischen Raum M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd nennen wir eine Folge (sequenceon[name.cvar.0]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonname.cvar.0n eine Cauchyfolge, falls es für jedes ϵ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanϵ ein n0http://mathhub.info/smglom/sets/set.omdoc?set?insetOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number gibt, so dass (d)<ϵhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthandϵ für alle n,mn0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?mmethannmOMFOREIGNscala.xml.Node$@6040b6ef.


  • Cousin-Primzahlen en

    Cousin-Primzahlen sind Primzahlen, die um 4 differieren.


  • Cullen-Zahl Show Notations en

    Eine Cullen-Zahl ist eine natürliche Zahl der Form (Cn)=((n(2n))+1)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/cullennumber.omdoc?cullennumber?Cullen-numbernhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn.


  • Cullen-Zahlen der zweiten Art

    Eine Woodall-Zahl ist eine natürliche Zahl der Form

    (Wn)=((n2n)-1)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/woodallnumber.omdoc?woodallnumber?Woodall-numbernhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn

    mit einer natürlichen Zahl nn. Woodall-Zahlen werden manchmal auch als Cullen-Zahlen der zweiten Art bezeichnet.

    Woodall-Zahlen die Primzahlen sind werden als Woodall-Primzahlen bezeichnet.


  • Definiendum en

    Wir nennen ein Paar a:=Ahttp://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationaA eine a definitionale Gleichung mit Definiendum aa und Definiens AA, wenn aa ein neues Symbol ist, das nicht in AA vorkommt.


  • Definiens en

    Wir nennen ein Paar a:=Ahttp://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationaA eine a definitionale Gleichung mit Definiendum aa und Definiens AA, wenn aa ein neues Symbol ist, das nicht in AA vorkommt.


  • Definitheit en

    Für eine Mente MM nennen wir eine Funktion d:(M×M)http://mathhub.info/smglom/sets/functions.omdoc?functions?fundhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsMMhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number eine Abstandsfunktion (or Metrik ) auf MM, falls die folgenden drei Eigenschaften für alle x,y,zMhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetxyzM gelten:

    1. 1.

      (d)=0http://mathhub.info/smglom/mv/equal.omdoc?equal?equald gdw. x=yhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalxy (Definitheit),

    2. 2.

      (d)=(d)http://mathhub.info/smglom/mv/equal.omdoc?equal?equaldd (Symmetrie) und

    3. 3.

      (d)((d)+(d))http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethandhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additiondd (Dreiecksungleichung).

    Wir nennen M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd einen metrischen Raum mit Grundmenge MM und Metrik dd.


  • definitionale Gleichung Show Notations en

    Wir nennen ein Paar a:=Ahttp://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationaA eine a definitionale Gleichung mit Definiendum aa und Definiens AA, wenn aa ein neues Symbol ist, das nicht in AA vorkommt.


  • dekadische Logarithmus Show Notations en

    Der dekadische Logarithmus (oder Zehnerlogarithmus) ist der Logarithmus zur Basis 10.


  • Delannoy-Zahl Show Notations en

    Eine Delannoy-Zahl D(m,n)http://mathhub.info/smglom/numbers/delannoynumber.omdoc?delannoynumber?Delannoy-numbermn ist die Anzahl der Wege von der südwest-Ecke 0,0http://mathhub.info/smglom/sets/pair.omdoc?pair?pair eines rechteckigen Gitters zur nordost-Ecke m,nhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairmn, unter Verwendung von Einzelschritten in Richtung Nord, Nordost oder Ost.

    Es folgt (multi-relation-expressionD(0,n)equalD(m,0)equal1)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numbers/delannoynumber.omdoc?delannoynumber?Delannoy-numbernhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/delannoynumber.omdoc?delannoynumber?Delannoy-numbermhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal und

    (D(m,n))=((D((m-1),n))+(D((m-1),(n-1)))+(D(m,(n-1))))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/delannoynumber.omdoc?delannoynumber?Delannoy-numbermnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numbers/delannoynumber.omdoc?delannoynumber?Delannoy-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionmnhttp://mathhub.info/smglom/numbers/delannoynumber.omdoc?delannoynumber?Delannoy-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numbers/delannoynumber.omdoc?delannoynumber?Delannoy-numbermhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn für alle (mnotequalmn0)http://mathhub.info/smglom/mv/equal.omdoc?equal?mnotequalmn.


  • Differenz Show Notations ro bg tr

    Die Differenz A\Bhttp://mathhub.info/smglom/sets/setdiff.omdoc?setdiff?set-differenceAB von Mengen AA und BB ist definiert als (bsetst[x]x)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstxx.


  • differenzierbar en

    Ist f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN differenzierbar auf MM, so definieren wir die Ableitungsfunktion (Dxf):MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfxMN von ff nach xx als

    (fundefeq[fxa]Dxf(a))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfxahttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa

    Die Abhängigkeit von xx ist bleibt dieser Notation (Lagrange Notation) implizit. In Leibniz Notation schreiben wir die Ableitungsfunktion von ff nach xx als Dxfhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfx. In Eulers Notation schreiben wir Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa und in Newtonnotation Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa.

    Wir nennen f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN differenzierbar an einem Häufungspunkt pMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetpM, falls der Grenzwert

    d:=(limxp((dB)(dA)))http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationdhttp://mathhub.info/smglom/calculus/functionlimit.omdoc?functionlimit?limitpxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6ef

    existiert. Dieser Grenzwert heißt Ableitung von ff nach xx an der Stelle pp. Wir nennen ff differenzierbar auf MM, falls ff differenzierbar an jedem mMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetmM ist.


  • differenzierbar auf en

    Ist f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN differenzierbar auf MM, so definieren wir die Ableitungsfunktion (Dxf):MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfxMN von ff nach xx als

    (fundefeq[fxa]Dxf(a))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfxahttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa

    Die Abhängigkeit von xx ist bleibt dieser Notation (Lagrange Notation) implizit. In Leibniz Notation schreiben wir die Ableitungsfunktion von ff nach xx als Dxfhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfx. In Eulers Notation schreiben wir Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa und in Newtonnotation Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa.

    Wir nennen f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN differenzierbar an einem Häufungspunkt pMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetpM, falls der Grenzwert

    d:=(limxp((dB)(dA)))http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationdhttp://mathhub.info/smglom/calculus/functionlimit.omdoc?functionlimit?limitpxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6ef

    existiert. Dieser Grenzwert heißt Ableitung von ff nach xx an der Stelle pp. Wir nennen ff differenzierbar auf MM, falls ff differenzierbar an jedem mMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetmM ist.


  • Digraph

    Ein gerichteter Graph (auch Digraph) ist ein Paar V,Ehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE aus einer Menge VV und einer Menge E(V,V)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetEhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairVV geordneter Paare über VV. Wir nennen VV die Knoten und EE die Kanten (auch Bögen) von GG.


  • Diophantische Gleichung en

    Eine Diophantische Gleichung ist eine Polynomgleichung mit ganzzahligen Koeffizienten, deren Variablen nur ganzzahlige Werte annehmen dürfen.


  • disjunkt ro en tr

    Zwei Mengen AA und BB heißen disjunkt, falls (AB)=http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/sets/intersection.omdoc?intersection?intersectionABhttp://mathhub.info/smglom/sets/emptyset.omdoc?emptyset?eset.

    Eine Familie von Mengen heißt paarweise disjunkt, wenn je zwei Mengen disjunkt sind.


  • Division Show Notations

    Die arithmetischen Operationen sind Addition additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?addition, Subtraktion, subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction, Multiplikation multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplication, Division divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?division und Exponentiation.


  • Doppel-Mersenne-Primzahl en

    Eine Doppel-Mersenne-Primzahl ist eine Doppel-Mersenne-Zahl, die auch Primzahl ist.


  • Doppel-Mersenne-Zahl Show Notations en

    Eine Doppel-Mersenne-Zahl ist eine Mersenne-Zahl der Form

    (Mp)=((2((2p)-1))-1)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/doublemersennenumber.omdoc?doublemersennenumber?double-Mersenne-numberphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationp

    Dabei ist pp der Exponent einer Mersenne-Primzahl.


  • Dreiecksungleichung en

    Für eine Mente MM nennen wir eine Funktion d:(M×M)http://mathhub.info/smglom/sets/functions.omdoc?functions?fundhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsMMhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number eine Abstandsfunktion (or Metrik ) auf MM, falls die folgenden drei Eigenschaften für alle x,y,zMhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetxyzM gelten:

    1. 1.

      (d)=0http://mathhub.info/smglom/mv/equal.omdoc?equal?equald gdw. x=yhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalxy (Definitheit),

    2. 2.

      (d)=(d)http://mathhub.info/smglom/mv/equal.omdoc?equal?equaldd (Symmetrie) und

    3. 3.

      (d)((d)+(d))http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethandhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additiondd (Dreiecksungleichung).

    Wir nennen M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd einen metrischen Raum mit Grundmenge MM und Metrik dd.


  • dreizehnte Smarandache-Konstante Show Notations en

    Die dreizehnte Smarandache-Konstante http://mathhub.info/smglom/smglom/smarandacheconstant13.omdoc?smarandacheconstant13?thirteenth-smarandache-constant ist definiert als

    http://mathhub.info/smglom/smglom/smarandacheconstant13.omdoc?smarandacheconstant13?thirteenth-smarandache-constant

    Dabei ist S(n)http://mathhub.info/smglom/smglom/smarandachefunction.omdoc?smarandachefunction?smarandache-functionn die Smarandache-Funktion.

    Die Summe konvergiert.


  • dritte Ableitung en

    Wir nennen ff nn mal differeinzierbar in aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM, falls dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa als Grenzwert existiert. Analog nennen wir ff nn mal differenzierbar auf MM, falls dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx existiert und auf MM total ist.

    Wir nennen ff unendlich differenzierbar oder glatt auf aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM oder MM, wenn dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa beziehungsweise dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx für alle nhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number existieren.

    Für ein nhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number definieren wir die nnte Ableitung einer Funktion f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN durch

    (fundefeq[nfxa]dnfdxn|x=a)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqnfxahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa

    Die erste Ableitung d1dx1fhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtfx von ff ist Dxfhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfx, d2dx2fhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtfx ist die zweite Ableitung von ff, d3dx3fhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtfx die dritte Ableitung von ff, usw. In der Leibniz Notation wird die nnte Ableiguntsfunktion von ff als dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx geschrieben.


  • dritte Smarandache-Konstante Show Notations en

    Die dritte Smarandache-Konstante http://mathhub.info/smglom/smglom/smarandacheconstant3.omdoc?smarandacheconstant3?third-smarandache-constant ist definiert als

    http://mathhub.info/smglom/smglom/smarandacheconstant3.omdoc?smarandacheconstant3?third-smarandache-constant

    Dabei ist S(n)http://mathhub.info/smglom/smglom/smarandachefunction.omdoc?smarandachefunction?smarandache-functionn die Smarandache-Funktion.


  • Durchschnitt Show Notations tr ro en

    Sind AA und BB Mengen, so ist der Durchschnitt ABhttp://mathhub.info/smglom/sets/intersection.omdoc?intersection?intersectionAB von AA und BB gegeben als (bsetst[x]x)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstxx.


  • echte Obermenge Show Notations ro tr en

    Eine Menge AA ist eine echte Obermenge einer Menge BB (schreibe ABhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?supersetAB), wenn BAhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetBA.

    Eine Menge AA heißt echte Teilmenge einer Menge BB (schreibe ABhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetAB), wenn ABhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?proper-subsetAB aber ABhttp://mathhub.info/smglom/sets/set.omdoc?set?nsetequalAB.


  • echte Teilmenge Show Notations ro tr en

    Eine Menge AA ist eine echte Obermenge einer Menge BB (schreibe ABhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?supersetAB), wenn BAhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetBA.

    Eine Menge AA heißt echte Teilmenge einer Menge BB (schreibe ABhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetAB), wenn ABhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?proper-subsetAB aber ABhttp://mathhub.info/smglom/sets/set.omdoc?set?nsetequalAB.


  • echter Teiler en

    Wir nennen 1, 1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction, nn und nhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn triviale divisoren von nn. Ein Teiler von nn, der nicht trivial ist, heißt echter Teiler von nn.


  • Ecken en

    Ein Graph ist ein Paar V,Ehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE, so dass VV eine Menge ist und E(V,V)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetEhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairVV eine Teilmenge der Menge der Paare aus VV. Wir nennen VV die Ecken (oder Punkte, Knoten) and EE die Kanten (oder Linien, Pfeile) von GG.


  • Eckenanzahl Show Notations en

    Die Eckenanzahl (Anzahl der Ecken eines Graphen) GG wird seine Ordnung genannt, man schreibt |G|http://mathhub.info/smglom/graphs/graphnumberofvertices.omdoc?graphnumberofvertices?graphnumberofverticesG.


  • einfach en

    Ist G=(V,E)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalGhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE ein gerichteter Graph, so nennen wir

    • einen Pfad pp in GG zyklisch (auch einen Zykel oder eine Schleife), falls (start(p))=(end(p))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pstartphttp://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pendp.

    • einen Zykel (v0,,vn)http://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlivn einfach, wenn vivjhttp://mathhub.info/smglom/mv/equal.omdoc?equal?notequalOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6ef für alle i,j1nhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?mbetweeneeijn mit ijhttp://mathhub.info/smglom/mv/equal.omdoc?equal?notequalij.

    • GG azyklisch, wenn GG keinen Zykel enthält.


  • einfach normale Zahl zur Basis en

    xx ist eine einfach normale Zahl zur Basis bb, wenn ihre Ziffern in der Darstellung zur Basis bb gleich häufig auftreten.

    Eine reelle Zahl ist absolut nicht-normal oder absolut unnormal, wenn sie in keiner Basis einfach normal ist.


  • einfache Ordnung en

    Eine partielle Ordnung RR heißt totale Ordnung (or einfache Ordnung oder lineare Ordnung), falls abhttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?poleab oder bahttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?poleba für alle a,bAhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabA.

    Wir nennen eine Struktur S,polehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureShttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole aus einer Menge SS und einer totalen Ordnung polehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole eine total geordnete Menge.


  • Eins en

    Eine Struktur Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR heißt Ring, wenn Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR eine Abelsch e Gruppe ist (die additive Struktur), Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR ein Monoid (die multiplikative Struktur) ist, sowie Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR is a Ringoid.

    Wir nennen 0 die Null des Rings und entsprechend 1 die Eins.


  • elften Smarandache-Konstanten Show Notations

    Die elften Smarandache-Konstanten eleventh-smarandache-constanthttp://mathhub.info/smglom/smglom/smarandacheconstant11.omdoc?smarandacheconstant11?eleventh-smarandache-constant sind definiert als

    s11(α)http://mathhub.info/smglom/smglom/smarandacheconstant11.omdoc?smarandacheconstant11?eleventh-smarandache-constantα

    Dabei ist S(n)http://mathhub.info/smglom/smglom/smarandachefunction.omdoc?smarandachefunction?smarandache-functionn die Smarandache-Funktion.

    Die Summe konvergiert für alle reellen Zahlen α>1http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanα.


  • Endknoten en

    Ist G=(V,E)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalGhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE gerichteter Graph, so nennen wir einen Vektor (p=((v0,,vn)))(V(n+1))http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalphttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlivnhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?nCartSpaceVhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionn einen Pfad in GG wenn (vi-1,vi)Ehttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6efE für alle 1inhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?betweeneein mit n>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethann.

    • v0OMFOREIGNscala.xml.Node$@6040b6ef heißt der Startknoten von pp (schreibe start(p)http://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pstartp )

    • vnOMFOREIGNscala.xml.Node$@6040b6ef heißt der Endknoten von pp (schreibe end(p)http://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pendp )

    • nn heißt die Länge von pp (schreibe len(p)http://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?plenp )


  • endlich en

    Wir nennen eine Menge AA is endlich mit Kardinalität (#(A))http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityAhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, wenn es eine bijektive Funktion f:A({n|(n<(#(A)))})http://mathhub.info/smglom/sets/functions.omdoc?functions?funfAhttp://mathhub.info/smglom/sets/set.omdoc?set?rsetsthttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-numbernhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthannhttp://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityA gibt.

    Die Kardinalität einer Menge AA wird oft auch als #(A)http://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityA, #(A)http://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityA, #(A)http://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityA oder #(A)http://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityA geschrieben.


  • Endwert en

    Wir definieren das Produkt über eine Folge aiOMFOREIGNscala.xml.Node$@6040b6ef durch

    (i=nmai):=(defined-piecewise(
    0wenn(n=m)
    )
    (
    (an)sonst
    )
    )
    http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdfromtonmiOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnmhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionOMFOREIGNscala.xml.Node$@6040b6ef

    Die Variable ii wird der Laufindex oder Multiplikationsindex genannt, nn die untere (oder Startwert) und mm die obereGrenze (oder Endwert) des Produkts, zusammen bestimmen sie den Multiplikationsbereich.

    Gebräuchlich sind auch die fogenden Varianten des Produktoperators: φaihttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdCollφOMFOREIGNscala.xml.Node$@6040b6ef und iSaihttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdInCollSiOMFOREIGNscala.xml.Node$@6040b6ef. Der erste spezifiziert den Multiplikationsbereich durch eine Formel φφ über ii und der zweite gibt den Multiplikationsbereich direkt als eine Menge SS an.


  • Endwert en

    Summation ist iterierte Addition. Wir definieren die Summe über eine Folge aiOMFOREIGNscala.xml.Node$@6040b6ef durch

    (i=nmai):=(defined-piecewise(
    0wenn(n=m)
    )
    (
    (an)sonst
    )
    )
    http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/numberfields/sum.omdoc?sum?sumnmiOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnmhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionOMFOREIGNscala.xml.Node$@6040b6ef

    Die Variable ii wird der Laufindex oder Summationsindex genannt, nn die untere (oder Startwert) und mm die obereGrenze (oder Endwert) der Summe, zusammen bestimmen sie den Summationsbereich.

    Gebräuchlich sind auch die fogenden Varianten des Summenoperators: (SumCollφ[i]ai)http://mathhub.info/smglom/numberfields/sum.omdoc?sum?SumCollφiOMFOREIGNscala.xml.Node$@6040b6ef und iSaihttp://mathhub.info/smglom/numberfields/sum.omdoc?sum?SumInCollSiOMFOREIGNscala.xml.Node$@6040b6ef. Der erste spezifiziert den Summationsbereich durch eine Formel φφ über ii und der zweite gibt den Summationsbereich als eine Menge SS an.


  • Erdös-Borwein-Konstante Show Notations en

    Die Erdös-Borwein-Konstante ist die Summe der Kehrwerte der Mersenne-Zahlen.

    Ehttp://mathhub.info/smglom/smglom/erdoesborweinconstant.omdoc?erdoesborweinconstant?Erdoes-Borwein-constant

  • erste Ableitung en

    Wir nennen ff nn mal differeinzierbar in aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM, falls dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa als Grenzwert existiert. Analog nennen wir ff nn mal differenzierbar auf MM, falls dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx existiert und auf MM total ist.

    Wir nennen ff unendlich differenzierbar oder glatt auf aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM oder MM, wenn dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa beziehungsweise dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx für alle nhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number existieren.

    Für ein nhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number definieren wir die nnte Ableitung einer Funktion f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN durch

    (fundefeq[nfxa]dnfdxn|x=a)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqnfxahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa

    Die erste Ableitung d1dx1fhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtfx von ff ist Dxfhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfx, d2dx2fhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtfx ist die zweite Ableitung von ff, d3dx3fhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtfx die dritte Ableitung von ff, usw. In der Leibniz Notation wird die nnte Ableiguntsfunktion von ff als dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx geschrieben.


  • erste Skewes-Zahl Show Notations en

    Die zweite Skewes-Zahl http://mathhub.info/smglom/smglom/skewesnumber.omdoc?skewesnumber?second-Skewes-number ist eine Obergrenze bis zu der nicht immer (π(n))(Li(n))http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?NumberPrimeNumbernhttp://mathhub.info/smglom/smglom/logarithmicintegralbig.omdoc?logarithmicintegralbig?logarithmicintbign gilt, vorausgesetzt, die Riemann-Hypothese ist falsch.

    =(10(10(101000)))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/smglom/skewesnumber.omdoc?skewesnumber?second-Skewes-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiation

  • erste Tschebyschow-Funktion en

    Die erste Tschebyschow-Funktion ϑ(x)http://mathhub.info/smglom/smglom/chebyshevfunction.omdoc?chebyshevfunction?firstchebyfuncx oder θxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationθx ist die Summe der Logarithmen der Primzahlen bis xx.

    ϑ(x)http://mathhub.info/smglom/smglom/chebyshevfunction.omdoc?chebyshevfunction?firstchebyfuncx

    Die zweite Tschebyschow-Funktion ψ(x)http://mathhub.info/smglom/smglom/chebyshevfunction.omdoc?chebyshevfunction?secchebyfuncx ist die Summe der Logarithmen der Primzahlen über alle Primzahlpotenzen bis xx.

    ψ(x)http://mathhub.info/smglom/smglom/chebyshevfunction.omdoc?chebyshevfunction?secchebyfuncx

    Dabei ist ΛΛ die Mangoldt-Funktion.


  • Euklid-Zahlen Show Notations en

    Euklid-Zahlen sind ganze Zahlen der Form

    (En)=((pn#)+1)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/euclidnumber.omdoc?euclidnumber?Euclid-numbernhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationOMFOREIGNscala.xml.Node$@6040b6ef%23

    wobei pn#http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationOMFOREIGNscala.xml.Node$@6040b6ef%23 das Produkt der ersten nn Primzahlen ist.


  • Euler-Mascheroni-Konstante Show Notations

    Die Euler-Mascheroni-Konstante (auch Eulersche Konstante) ist eine mathematische Konstante, die mit γγ bezeichnet wird.

    γhttp://mathhub.info/smglom/smglom/eulermascheroni.omdoc?eulermascheroni?eulermascheroni
    γ=0.5772156649015329...http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?startswithhttp://mathhub.info/smglom/smglom/eulermascheroni.omdoc?eulermascheroni?eulermascheroni

  • Eulers Notation en

    Ist f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN differenzierbar auf MM, so definieren wir die Ableitungsfunktion (Dxf):MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfxMN von ff nach xx als

    (fundefeq[fxa]Dxf(a))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfxahttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa

    Die Abhängigkeit von xx ist bleibt dieser Notation (Lagrange Notation) implizit. In Leibniz Notation schreiben wir die Ableitungsfunktion von ff nach xx als Dxfhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfx. In Eulers Notation schreiben wir Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa und in Newtonnotation Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa.


  • Eulersche Konstante Show Notations

    Die Euler-Mascheroni-Konstante (auch Eulersche Konstante) ist eine mathematische Konstante, die mit γγ bezeichnet wird.

    γhttp://mathhub.info/smglom/smglom/eulermascheroni.omdoc?eulermascheroni?eulermascheroni
    γ=0.5772156649015329...http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?startswithhttp://mathhub.info/smglom/smglom/eulermascheroni.omdoc?eulermascheroni?eulermascheroni

  • Eulersche Zahl Show Notations en

    Die Eulersche Zahl ee (benannt nach dem Schweizer Mathematiker Leonhard Euler) ist eine mathematische Konstante.

    ehttp://mathhub.info/smglom/numberfields/eulersnumber.omdoc?eulersnumber?eulersnumber
    e=2.718281828459045...http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?startswithhttp://mathhub.info/smglom/numberfields/eulersnumber.omdoc?eulersnumber?eulersnumber

  • Exponentiation Show Notations

    Die arithmetischen Operationen sind Addition additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?addition, Subtraktion, subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction, Multiplikation multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplication, Division divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?division und Exponentiation.


  • Factorion en

    Ein Factorion ist eine natürliche Zahl, die gleich der Summe der Fakultäten ihrer Ziffern ist.

    Zum Beispiel

    (multi-relation-expression145equal1+24+120equal(1!)+(4!)+(5!))http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/factorial.omdoc?factorial?factorialhttp://mathhub.info/smglom/numberfields/factorial.omdoc?factorial?factorialhttp://mathhub.info/smglom/numberfields/factorial.omdoc?factorial?factorial

  • Faktorielle Show Notations en

    Die Fakultät nfactorialhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnhttp://mathhub.info/smglom/numberfields/factorial.omdoc?factorial?factorial (manchmal, besonders in Österreich, auch Faktorielle genannt) ist in der Mathematik eine Funktion, die einer natürlichen Zahl nn das Produkt aller natürlichen Zahlen kleiner und gleich dieser Zahl zuordnet.


  • Fakultät Show Notations en

    Die Fakultät nfactorialhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnhttp://mathhub.info/smglom/numberfields/factorial.omdoc?factorial?factorial (manchmal, besonders in Österreich, auch Faktorielle genannt) ist in der Mathematik eine Funktion, die einer natürlichen Zahl nn das Produkt aller natürlichen Zahlen kleiner und gleich dieser Zahl zuordnet.


  • Farey-Folge en

    Eine Farey-Folge ist eine geordnete Menge der gekürzten Brüche zwischen 0 und 1, deren jeweiliger Nenner den Index nn nicht übersteigt. Jede Farey-Folge beginnt mit 0, dargestellt durch 01http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?division, und endet mit 1, dargestellt durch 11http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?division.

    Die ersten Farey-Folgen sind

    (F1)=({(01),(11)})http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/smglom/fareysequence.omdoc?fareysequence?fareyseqhttp://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?division

    (F2)=({(01),(12),(11)})http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/smglom/fareysequence.omdoc?fareysequence?fareyseqhttp://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?division

    (F3)=({(01),(13),(12),(23),(11)})http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/smglom/fareysequence.omdoc?fareysequence?fareyseqhttp://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?division

    (F4)=({(01),(14),(13),(12),(23),(34),(11)})http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/smglom/fareysequence.omdoc?fareysequence?fareyseqhttp://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?division


  • Fastprimzahl en

    Eine Halbprimzahl oder auch Fastprimzahl ist eine natürliche Zahl, die das Produkt aus zwei (nicht notwendigerweise verschiedenen) Primzahlen ist.


  • Fermatsche Primzahl Show Notations en

    Eine Fermatsche Primzahl ist eine Primzahl der Form

    (Fn)=((2(2n))+1)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/primes/fermatprime.omdoc?fermatprime?Fermat-primenhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn

  • Fermatsche Zahl Show Notations en

    Eine Fermatsche Zahl ist eine natürliche Zahl der Form (fundefeq[n]Fn)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqnhttp://mathhub.info/smglom/numbers/fermatnumber.omdoc?fermatnumber?Fermat-numbern.


  • Feynman-Punkt en

    Der Feynman-Punkt ist ein Bereich von 6 aufeinanderfolgenden Neunen in der Dezimaldarstellung der Zahl πhttp://mathhub.info/smglom/numberfields/pinumber.omdoc?pinumber?pinumber, beginnend mit der 762. Dezimalstelle.


  • Fibonacci Zahlen Show Notations en

    Die Fibonacci Zahlen werden rekursiv definiert:

    (Fn)=(defined-piecewise(
    0wenn(n=0)
    )
    (
    1wenn(n=1)
    )
    (
    ((F(n-1))+(F(n-2)))sonst
    )
    )
    http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/fibonaccinumbers.omdoc?fibonaccinumbers?Fibonacci-numbersnhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numbers/fibonaccinumbers.omdoc?fibonaccinumbers?Fibonacci-numbershttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numbers/fibonaccinumbers.omdoc?fibonaccinumbers?Fibonacci-numbershttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn

  • Fibonacci-Polynome en

    Die Fibonacci-Polynome werden rekursiv definiert

    ((Fn(x))x)=(defined-piecewise(
    0wenn(n=0)
    )
    (
    1wenn(n=1)
    )
    (
    ((x(F(n-1)(x))x)+((F(n-2)(x))x))wenn(n2)
    )
    )
    http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/fibonaccipolynomials.omdoc?fibonaccipolynomials?fibpolnxhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationxhttp://mathhub.info/smglom/smglom/fibonaccipolynomials.omdoc?fibonaccipolynomials?fibpolhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/fibonaccipolynomials.omdoc?fibonaccipolynomials?fibpolhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnxhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methann

    Die ersten Fibonacci-Polynome sind:

    ((F0(x))x)=0http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/fibonaccipolynomials.omdoc?fibonaccipolynomials?fibpolx

    ((F1(x))x)=1http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/fibonaccipolynomials.omdoc?fibonaccipolynomials?fibpolx

    ((F2(x))x)=xhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/fibonaccipolynomials.omdoc?fibonaccipolynomials?fibpolxx

    ((F3(x))x)=((x2)+1)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/fibonaccipolynomials.omdoc?fibonaccipolynomials?fibpolxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationx

    ((F4(x))x)=((x3)+(2x))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/fibonaccipolynomials.omdoc?fibonaccipolynomials?fibpolxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationx

    ((F5(x))x)=((x4)+(3(x2))+1)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/fibonaccipolynomials.omdoc?fibonaccipolynomials?fibpolxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationx


  • Fortunate-Zahl en

  • Franel-Zahlen Show Notations en

    Die Franel-Zahlen Frnhttp://mathhub.info/smglom/numbers/franelnumber.omdoc?franelnumber?Franel-numbern sind die ganzenZahlen

    Frnhttp://mathhub.info/smglom/numbers/franelnumber.omdoc?franelnumber?Franel-numbern

  • Franel-Zahlen en

  • freundliches -Tupel en

    nn natürliche Zahlen mit der gleichen Abundancy bilden ein freundliches nn-Tupel.


  • Funktion en

    Wenn wir nicht festlegen wollen, ob eine partielle Funktion total ist, sprechen wir einfach von einer Funktion.


  • fünfte Smarandache-Konstante Show Notations en

    Die fünfte Smarandache-Konstante http://mathhub.info/smglom/smglom/smarandacheconstant5.omdoc?smarandacheconstant5?fifth-smarandache-constant ist definiert als

    http://mathhub.info/smglom/smglom/smarandacheconstant5.omdoc?smarandacheconstant5?fifth-smarandache-constant

    Dabei ist S(n)http://mathhub.info/smglom/smglom/smarandachefunction.omdoc?smarandachefunction?smarandache-functionn die Smarandache-Funktion.

    Die Summe konvergiert.


  • fünfzehnte Smarandache-Konstante en

    Die fünfzehnte Smarandache-Konstante http://mathhub.info/smglom/smglom/smarandacheconstant15.omdoc?smarandacheconstant15?fifteenth-smarandache-constant ist definiert als

    http://mathhub.info/smglom/smglom/smarandacheconstant15.omdoc?smarandacheconstant15?fifteenth-smarandache-constant

    Dabei ist S(n)http://mathhub.info/smglom/smglom/smarandachefunction.omdoc?smarandachefunction?smarandache-functionn die Smarandache-Funktion.

    Die Summe konvergiert.


  • ganze Teil Show Notations en

    Die Abrundung (oder der ganze Teil)

    r
    http://mathhub.info/smglom/numberfields/ceilingfloor.omdoc?ceilingfloor?floorr einer reellen Zahl rr ist die größte ganze Zahl, die nicht größer ist als rr. Die Aufrundungsfunktion wird auch die Gaussklammer genannt; dann wird sie als r
    http://mathhub.info/smglom/numberfields/ceilingfloor.omdoc?ceilingfloor?floorr geschrieben.


  • ganze Zahl en

    Die Menge http://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?integers der ganzen Zahlen ist {(,(2),(1),0,1,2,)}http://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?dotsaseqdotshttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction. Ein Element von http://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?integers wird als ganze Zahl bezeichnet.

    Die Menge der negativen ganzen Zahlen http://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?negative-integers ist {(,(3),(2),(1))}http://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?dotsaseqhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction.


  • ganzen Zahlen Show Notations en

    Die Menge http://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?integers der ganzen Zahlen ist {(,(2),(1),0,1,2,)}http://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?dotsaseqdotshttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction. Ein Element von http://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?integers wird als ganze Zahl bezeichnet.

    Die Menge der negativen ganzen Zahlen http://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?negative-integers ist {(,(3),(2),(1))}http://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?dotsaseqhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction.


  • ganzzahlige Intervall Show Notations en

    Wir definieren ein ganzzahlige Intervall als eine Menge konsekutiver ganzer Zahlen: (fundefeq[ab][a,b])http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqabhttp://mathhub.info/smglom/calculus/interval.omdoc?interval?integer-intervalab


  • Gaussklammer Show Notations en

    Die Abrundung (oder der ganze Teil)

    r
    http://mathhub.info/smglom/numberfields/ceilingfloor.omdoc?ceilingfloor?floorr einer reellen Zahl rr ist die größte ganze Zahl, die nicht größer ist als rr. Die Aufrundungsfunktion wird auch die Gaussklammer genannt; dann wird sie als r
    http://mathhub.info/smglom/numberfields/ceilingfloor.omdoc?ceilingfloor?floorr geschrieben.


  • Gelfond-Schneider Konstante en

    Die Gelfond-Schneider Konstante oder Hilbert-Zahl ist die folgende reelle Zahl

    (2(2))=(2.665144142690225)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?square-roothttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplication

  • Gelfond’s Konstante en

    Gelfond’s Konstante ist die folgende reelleZahl

    (eπ)=23.14069263277927...http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?startswithhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/eulersnumber.omdoc?eulersnumber?eulersnumberhttp://mathhub.info/smglom/numberfields/pinumber.omdoc?pinumber?pinumber

  • geometrische Mittel Show Notations en

    Das geometrische Mittel einer Menge M=({(a1,,an)})http://mathhub.info/smglom/mv/equal.omdoc?equal?equalMhttp://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlian ist definiert als:

    ((Gequal)((i=1nai)(1n)))=((a1a2an)n)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/geometricmean.omdoc?geometricmean?geometric-meanhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdfromtoniOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?rootnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6ef

  • geordnete Menge en

    Wir nennen eine Struktur S,polehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureShttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole aus einer Menge SS und einer partiellen Ordnung polehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole eine geordnete Menge.

    Eine partielle Ordnung RR heißt totale Ordnung (or einfache Ordnung oder lineare Ordnung), falls abhttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?poleab oder bahttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?poleba für alle a,bAhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabA.

    Wir nennen eine Struktur S,polehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureShttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole aus einer Menge SS und einer totalen Ordnung polehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole eine total geordnete Menge.


  • gerichteter Graph en

    Ein gerichteter Graph (auch Digraph) ist ein Paar V,Ehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE aus einer Menge VV und einer Menge E(V,V)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetEhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairVV geordneter Paare über VV. Wir nennen VV die Knoten und EE die Kanten (auch Bögen) von GG.


  • geschlossen . en

    Ein topologischer Raum X,Ohttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureXOMFOREIGNscala.xml.Node$@6040b6ef ist eine Menge XX zusammen mit einer Familie O(𝒫(X))http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/sets/powerset.omdoc?powerset?powersetX, so daß

    1. 1.

      ,XOhttp://mathhub.info/smglom/sets/set.omdoc?set?minsethttp://mathhub.info/smglom/sets/emptyset.omdoc?emptyset?esetXOMFOREIGNscala.xml.Node$@6040b6ef.

    2. 2.

      (SSs)Ohttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/union.omdoc?union?munionCollectionOMFOREIGNscala.xml.Node$@6040b6efSsOMFOREIGNscala.xml.Node$@6040b6ef falls SShttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetSOMFOREIGNscala.xml.Node$@6040b6ef.

    3. 3.

      (SSs)Ohttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/intersection.omdoc?intersection?mintersectCollectionOMFOREIGNscala.xml.Node$@6040b6efSsOMFOREIGNscala.xml.Node$@6040b6ef falls SShttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetSOMFOREIGNscala.xml.Node$@6040b6ef endlich.

    Dann heißt OOMFOREIGNscala.xml.Node$@6040b6ef eine Topologie) auf XX. Elemente der Topologie OOMFOREIGNscala.xml.Node$@6040b6ef heißen offene Mengen und ihre Komplemente abgeschlossen oder einfach geschlossen . Eine Teilmenge von XX kann weder gesclossen noch offen, oder gesclossen, oder offen oder beides.


  • geschlossene Kugel Show Notations en

    In einem metrischen Raum M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd nennen wir die Menge (fundefeq[rx]𝐵(r,x))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqrxhttp://mathhub.info/smglom/calculus/ball.omdoc?ball?open-ballrx die offene Kugel nd (fundefeq[rx]B¯(r,x))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqrxhttp://mathhub.info/smglom/calculus/ball.omdoc?ball?closed-ballrx die geschlossene Kugel um xx mit Radius rr. Wir schreiben auch 𝐵(r,x)http://mathhub.info/smglom/calculus/ball.omdoc?ball?open-ballrx und B¯(r,x)http://mathhub.info/smglom/calculus/ball.omdoc?ball?closed-ballrx.


  • glatt en

    Wir nennen ff nn mal differeinzierbar in aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM, falls dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa als Grenzwert existiert. Analog nennen wir ff nn mal differenzierbar auf MM, falls dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx existiert und auf MM total ist.

    Wir nennen ff unendlich differenzierbar oder glatt auf aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM oder MM, wenn dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa beziehungsweise dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx für alle nhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number existieren.


  • gleich Show Notations tr ro ru

    Zwei Mengen AA and BB sind gleich (written ABhttp://mathhub.info/smglom/sets/set.omdoc?set?setequalAB), wenn sie die gleichen Elemente haben.


  • gleich Show Notations en

    Wir nennen zwei mathematische Objekte aa and bb gleich, (schreibe a=bhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalab), wenn es keine Eigenschaften gibt, die sie auseinanderhalten.


  • glücklichen Zahlen von Euler en

    Die glücklichen Zahlen von Euler sind positive ganze Zahlen nn, für die ((x2)-x)+nhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationxxn für x=0=1==(n-1)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn eine Primzahl ist.


  • Googol en

    Googol ist die große Zahl 10100http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiation.


  • Googolminex en

    Googolminex ist der Kehrwert von Googolplex.


  • Googolplex en

    Googolplexplex ist die Zahl

    (Googolplexplex)=(10(Googolplex))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationGoogolplexplexhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationGoogolplex

    Googolplexplexplex ist die Zahl

    (Googolplexplexplex)=(10(Googolplexplex))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationGoogolplexplexplexhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationGoogolplexplex

    Googolplex ist die Zahl

    (multi-relation-expressionGoogolplexequal10(googol)equal10(10100))http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationGoogolplexhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationgoogolhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiation

  • Googolplexian en

    Googolplexian ist die Zahl

    (multi-relation-expressionGoogolplexianequalGoogolplexplexequal10(Googolplex))http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationGoogolplexianhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationGoogolplexplexhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationGoogolplex

  • Googolplexplex

    Googolplexplex ist die Zahl

    (Googolplexplex)=(10(Googolplex))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationGoogolplexplexhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationGoogolplex

    Googolplexplexplex ist die Zahl

    (Googolplexplexplex)=(10(Googolplexplex))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationGoogolplexplexplexhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationGoogolplexplex

  • Googolplexplexplex

    Googolplexplexplex ist die Zahl

    (Googolplexplexplex)=(10(Googolplexplex))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationGoogolplexplexplexhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationGoogolplexplex

  • Graph en

    Ein Graph ist ein Paar V,Ehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE, so dass VV eine Menge ist und E(V,V)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetEhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairVV eine Teilmenge der Menge der Paare aus VV. Wir nennen VV die Ecken (oder Punkte, Knoten) and EE die Kanten (oder Linien, Pfeile) von GG.


  • Graph en

    Der Graph einer Funktion ff ist die Menge aller Paare x,(fx)http://mathhub.info/smglom/sets/pair.omdoc?pair?pairxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationfx.


  • Gregory-Zahl en

    Eine Gregory-Zahl ist eine reelle Zahl der Form

    Gxhttp://mathhub.info/smglom/smglom/gregorynumber.omdoc?gregorynumber?Gergory-numberx

    Dabei ist xx eine rationale Zahl größer oder gleich 1.


  • Grenzwert Show Notations en

    Seien A,dAhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureAOMFOREIGNscala.xml.Node$@6040b6ef und B,dBhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureBOMFOREIGNscala.xml.Node$@6040b6ef metrische Räume und f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN für MAhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetMA und NBhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetNB, dann nennen wir lMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetlM en Grenzwert (oder Limes) von ff wenn xx gegen einen Häufungspunkt pNhttp://mathhub.info/smglom/sets/set.omdoc?set?insetpN strebt (schreibe limxp(f)http://mathhub.info/smglom/calculus/functionlimit.omdoc?functionlimit?limitpxf), falls es für jedes ϵ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanϵ ein δ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanδ gibt, so dass für alle xx mit (multi-relation-expression0lethan(dB)lethanδ)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanδ auch (dA)ϵhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanOMFOREIGNscala.xml.Node$@6040b6efϵ gilt.


  • Grenzwert Show Notations

    Ist M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd ein metrischer Raum und (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan eine Folge mit (ai)Mhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselaiM für alle ihttp://mathhub.info/smglom/sets/set.omdoc?set?insetihttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, so sagen wir daß (sequenceon[name.cvar.0]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonname.cvar.0n gegen gMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetgM konvergiert (wir nennen gg den Grenzwert oder limitLimes und schreiben limnanhttp://mathhub.info/smglom/calculus/sequenceConvergence.omdoc?sequenceConvergence?limitnOMFOREIGNscala.xml.Node$@6040b6ef ), falls es für jedes ϵ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanϵ ein n0http://mathhub.info/smglom/sets/set.omdoc?set?insetOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, so daß (d)<ϵhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthandϵ für alle nn0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methannOMFOREIGNscala.xml.Node$@6040b6ef.


  • Grundmenge en

    Für eine Mente MM nennen wir eine Funktion d:(M×M)http://mathhub.info/smglom/sets/functions.omdoc?functions?fundhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsMMhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number eine Abstandsfunktion (or Metrik ) auf MM, falls die folgenden drei Eigenschaften für alle x,y,zMhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetxyzM gelten:

    1. 1.

      (d)=0http://mathhub.info/smglom/mv/equal.omdoc?equal?equald gdw. x=yhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalxy (Definitheit),

    2. 2.

      (d)=(d)http://mathhub.info/smglom/mv/equal.omdoc?equal?equaldd (Symmetrie) und

    3. 3.

      (d)((d)+(d))http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethandhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additiondd (Dreiecksungleichung).

    Wir nennen M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd einen metrischen Raum mit Grundmenge MM und Metrik dd.


  • Gruppe en

    Ein Magma Ghttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureG heißt Gruppe, wenn (aa)=(bb)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionaahttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionbb, (a(bb))=ahttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionahttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionbba, ((aa)(bc))=(cb)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionaahttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionbchttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisioncb und ((ac)(bc))=(ab)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionachttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionbchttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionab für alle a,b,cGhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabcG.


  • größte gemeinsame Teiler en

    Der größte gemeinsame Teiler von zwei oder mehr ganzen Zahlen, von denen mindestens eine nicht Null ist, ist die größte natürliche Zahl, die alle Zahlen ohne Rest teilt.


  • gut en

    Ein Binomialkoeffizient 𝒞knhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficientnk mit k2http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methank ist gut, wenn für seinen kleinsten Primfaktor (leastprimefactorhttp://mathhub.info/smglom/smglom/leastprimefactor.omdoc?leastprimefactor?leastprimefactor) gilt

    (lpf((𝒞kn)))>khttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanhttp://mathhub.info/smglom/smglom/leastprimefactor.omdoc?leastprimefactor?leastprimefactorhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficientnkk

  • gute Primzahl en

    Die nn-te Primzahl pnhttp://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?prime-numbern heißt gute Primzahl, falls für alle Paare von Primzahlen p(n-i)http://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?prime-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionni und p(n+i)http://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?prime-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionni, wobei ii von 1 bis n-1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn geht, gilt:

    ((pn)2)>(p(n-i)p(n+i))http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?prime-numbernhttp://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?prime-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnihttp://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?prime-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionni

  • Halbprimzahl en

    Eine Halbprimzahl oder auch Fastprimzahl ist eine natürliche Zahl, die das Produkt aus zwei (nicht notwendigerweise verschiedenen) Primzahlen ist.


  • Halbring en

    Ein Halbring ist ein Ringoid Shttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureS, so dass Shttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureS und Shttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureS Monoide sind und additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?addition kommutativ.


  • Hardy-Ramanujan-Zahl en

    Die Hardy-Ramanujan-Zahl 1729 ist die kleinste natürliche Zahl, für die es genau zwei Darstellungen als Summe zweier Kubikzahlen gibt.

    (multi-relation-expression1729equal(13)+(123)equal(93)+(103))http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiation

  • harmonische Mittel Show Notations en

    Das harmonische Mittel der reellen Zahlen M=({(x1,,xn)})http://mathhub.info/smglom/mv/equal.omdoc?equal?equalMhttp://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlixn ist definiert als

    (multi-relation-expressionHMequaln((1x1)+(1x2)++(1xn))equaln(i=1n(1xi)))http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/harmonicmean.omdoc?harmonicmean?harmonic-meanMhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionnhttp://mathhub.info/smglom/numberfields/sum.omdoc?sum?sumnihttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionOMFOREIGNscala.xml.Node$@6040b6ef

  • harmonische Reihe en

    Die harmonische Reihe ist die divergente unendliche Reihe

    (infinite-sum1[n]1n)http://mathhub.info/smglom/calculus/infinitesum.omdoc?infinitesum?infinite-sumnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionn

  • harmonische-Teiler-Zahl en

    Eine harmonische-Teiler-Zahl, oder Ore-Zahl, ist eine positive ganze Zahl, deren Harmonisches Mittel ihrer Teiler eine ganze Zahl ist.


  • Harshad-Zahl en

    Eine Harshad-Zahl oder Niven-Zahl in einer gegebenen Zahlenbasis ist eine ganze Zahl die durch die Summe ihrer Ziffern in dieser Zahlenbasis teilbar ist.


  • Hausdorff-Grenzwert Show Notations en

    Ist X,Ohttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureXOMFOREIGNscala.xml.Node$@6040b6ef ein Hausdorff-Raum, so sagen wir dass eine Folge (sequenceon[name.cvar.0]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonname.cvar.0n gegen xXhttp://mathhub.info/smglom/sets/set.omdoc?set?insetxX konvergiert (wir nennen xx den Hausdorff-Grenzwert und schreiben xx), falls jede Umgebung von xx nur endlich viele xiOMFOREIGNscala.xml.Node$@6040b6ef enthält.


  • Hilbert-Primzahl en

    Eine Hilbert-Primzahl ist eine Hilbert-Zahl, die nicht durch eine kleinere Hilbert-Zahl (außer der 1) teilbar ist.


  • Hilbert-Zahl en

    Die Gelfond-Schneider Konstante oder Hilbert-Zahl ist die folgende reelle Zahl

    (2(2))=(2.665144142690225)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?square-roothttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplication

  • hochzusammengesetzte Zahl Show Notations en

    Eine hochzusammengesetzte Zahl ist eine positive ganze Zahl, die mehr Teiler besitzt als jede kleinere positive ganze Zahl.


  • hyperbolische Integralkosinus en

    Der hyperbolische Integralkosinus Chi(x)http://mathhub.info/smglom/smglom/hyperboliccosineintegral.omdoc?hyperboliccosineintegral?hyperboliccosineintx oder Chi(x)http://mathhub.info/smglom/smglom/hyperboliccosineintegral.omdoc?hyperboliccosineintegral?hyperboliccosineintx wird definiert durch

    Chi(x)http://mathhub.info/smglom/smglom/hyperboliccosineintegral.omdoc?hyperboliccosineintegral?hyperboliccosineintx

  • hyperbolische Integralsinus en

    Der hyperbolische Integralsinus Shi(x)http://mathhub.info/smglom/smglom/hyperbolicsineintegral.omdoc?hyperbolicsineintegral?hyperbolicsineintx oder Shi(x)http://mathhub.info/smglom/smglom/hyperbolicsineintegral.omdoc?hyperbolicsineintegral?hyperbolicsineintx wird definiert durch

    Shi(x)http://mathhub.info/smglom/smglom/hyperbolicsineintegral.omdoc?hyperbolicsineintegral?hyperbolicsineintx

  • Häufungspunkt en

    Sei SS eine Teilmenge eines topologischen Raums XX. Dann nennen wir xXhttp://mathhub.info/smglom/sets/set.omdoc?set?insetxX einen Häufungspunkt von SS, wenn jede Umgebung von xx mindestens einen Punkt sShttp://mathhub.info/smglom/sets/set.omdoc?set?insetsS enthält mit xshttp://mathhub.info/smglom/mv/equal.omdoc?equal?notequalxs.


  • höfliche Zahl en

    Eine höfliche Zahl ist eine positive ganze Zahl, die als Summe von zwei oder mehr aufeinanderfolgenden positiven ganzen Zahlen dargestellt werden kann.


  • Identitätsfunktion Show Notations en

    Für eine Menge AA bildet die Identitätsfunktion (IdA):AAhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funhttp://mathhub.info/smglom/sets/identity-function.omdoc?identity-function?identity-functionAAA auf AA jedes aAhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaA auf sich selbst ab.


  • Idoneal-Zahl

    Eine positive ganze Zahl nn ist eine Idoneal-Zahl genau dann, wenn sie nicht als (ab)+(bc)+(ac)http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationabhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationbchttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationac mit verschiedenen positiven ganzen Zahlen aa, bb, und cc geschrieben werden kann.


  • imaginäre Einheit Show Notations en

    Die Menge http://mathhub.info/smglom/numberfields/complexnumbers.omdoc?complexnumbers?complex-number der komplexen Zahlen besteht aus Zahlen der Form a+(b𝑖)http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionahttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationbhttp://mathhub.info/smglom/numberfields/complexnumbers.omdoc?complexnumbers?imaginary-unit, mit a,bhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number. Wir nennen 𝑖http://mathhub.info/smglom/numberfields/complexnumbers.omdoc?complexnumbers?imaginary-unit imaginäre Einheit.


  • infimum Show Notations

    Sei S,polehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureShttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole eine geordnete Menge und TShttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetTS, dann nennen wir die kleinste obere Schranke sup(T)http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?supremumT (größte untere Schranke inf(T)http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?infimumT) von TT das Supremum (Infimum) von TT (falls dies existiert).

    Ist ee ein Ausdruck und φφ eine Bedingung (in einer Variablen xx), so schreiben wir (bsupremumφ[x]e)http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?bsupremumφxe für sup((bsetst[x]φ))http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?supremumhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstxφ und nennen es das supremum für ee über φφ. Analog schreiben wir (binfimumφ[x]e)http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?binfimumφxe für inf((bsetst[x]φ))http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?infimumhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstxφ und nennen es das infimum für ee über φφ.


  • Infimum Show Notations en

    Sei S,polehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureShttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole eine geordnete Menge und TShttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetTS, dann nennen wir die kleinste obere Schranke sup(T)http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?supremumT (größte untere Schranke inf(T)http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?infimumT) von TT das Supremum (Infimum) von TT (falls dies existiert).

    Ist ee ein Ausdruck und φφ eine Bedingung (in einer Variablen xx), so schreiben wir (bsupremumφ[x]e)http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?bsupremumφxe für sup((bsetst[x]φ))http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?supremumhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstxφ und nennen es das supremum für ee über φφ. Analog schreiben wir (binfimumφ[x]e)http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?binfimumφxe für inf((bsetst[x]φ))http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?infimumhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstxφ und nennen es das infimum für ee über φφ.


  • initial

    Sei G=(V,E)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalGhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE ein gerichteter Graph, dann nennen wir einen Knoten vVhttp://mathhub.info/smglom/sets/set.omdoc?set?insetvV

    • initial (oder eine Quelle) in GG, wenn es kein wVhttp://mathhub.info/smglom/sets/set.omdoc?set?insetwV gibt, so dass (w,v)Ehttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairwvE.

    • terminal (oder Senke) in GG, wenn es kein wVhttp://mathhub.info/smglom/sets/set.omdoc?set?insetwV gibt, so dass (v,w)Ehttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairvwE.


  • injektiv en

    Eine Funktion f:SThttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfST heißt injektiv, wenn für alle x,yShttp://mathhub.info/smglom/sets/set.omdoc?set?minsetxyS mit (f)=(f)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalff gilt x=yhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalxy.


  • Integralkosinus Show Notations en

    Der Integralkosinus Cin(x)http://mathhub.info/smglom/smglom/cosineintegralint.omdoc?cosineintegralint?cosine-integralx wird definiert durch

    Cin(x)http://mathhub.info/smglom/smglom/cosineintegralint.omdoc?cosineintegralint?cosine-integralx

  • Integrallogarithmus en

    Der Integrallogarithmus Li(x)http://mathhub.info/smglom/smglom/logarithmicintegralbig.omdoc?logarithmicintegralbig?logarithmicintbigx ist für alle positiven reellen Zahlen x>1http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanx definiert:

    Li(x)http://mathhub.info/smglom/smglom/logarithmicintegralbig.omdoc?logarithmicintegralbig?logarithmicintbigx

  • inverse Function Show Notations en

  • irrationale Zahl en

    Eine irrationale Zahl ist eine reelle Zahl, aber keine rationale Zahl.


  • irreflexiv en

    Eine Relation R(A×A)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetRhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAA heißt

    • reflexiv auf AA, wenn (a,a)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairaaR für alle aAhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaA, und

    • irreflexiv (or antireflexiv) auf AA, wenn (a,a)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?ninsethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairaaR für alle aAhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaA.


  • iterierte Querprodukt en

    Die Anzahl der notwendigen Schritte, um durch wiederholte Querprodukte zu einer einstelligen Zahl zu gelangen, nennt man Beharrlichkeit der Zahl.

    Das iterierte Querprodukt einer Zahl erhält man, indem man von dem Querprodukt dieser Zahl so lange wieder das Querprodukt bildet, bis nur noch eine einstellige Zahl übrig bleibt.


  • iterierte Quersumme

    Die iterierte Quersumme einer Zahl erhält man, indem man von der Quersumme dieser Zahl so lange wieder die Quersumme bildet, bis nur noch eine einstellige Zahl übrig bleibt.


  • Jacobi-Madden-Gleichung

    Die Jacobi-Madden-Gleichung ist die DiophantischeGleichung

    ((a4)+(b4)+(c4)+(d4))=((a+b+c+d)4)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationahttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationbhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationchttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationdhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionabcd

  • Jacobsthal-Lucas-Zahlen w

    Die Jacobsthal-Lucas-Zahlen werden rekursiv definiert:

    (Ln)=(defined-piecewise(
    2wenn(n=0)
    )
    (
    1wenn(n=1)
    )
    (
    ((L(n-1))+(2(L(n-2))))sonst
    )
    )
    http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/jacobsthallucasnumbers.omdoc?jacobsthallucasnumbers?Jacobsthal-Lucas-numbernhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numbers/jacobsthallucasnumbers.omdoc?jacobsthallucasnumbers?Jacobsthal-Lucas-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numbers/jacobsthallucasnumbers.omdoc?jacobsthallucasnumbers?Jacobsthal-Lucas-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn

    Definition in geschlossener Form: (Ln)=((2n)+((1)n))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/jacobsthallucasnumbers.omdoc?jacobsthallucasnumbers?Jacobsthal-Lucas-numbernhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn


  • Jacobsthal-Zahlen en

    Die Jacobsthal-Zahlen werden rekursiv definiert:

    J0http://mathhub.info/smglom/numbers/jacobsthalnumbers.omdoc?jacobsthalnumbers?jacobsthalnumbers equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal 0
    J1http://mathhub.info/smglom/numbers/jacobsthalnumbers.omdoc?jacobsthalnumbers?jacobsthalnumbers equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal 1
    Jnhttp://mathhub.info/smglom/numbers/jacobsthalnumbers.omdoc?jacobsthalnumbers?jacobsthalnumbersn equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal (J(n-1))+(2(J(n-2)))http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numbers/jacobsthalnumbers.omdoc?jacobsthalnumbers?jacobsthalnumbershttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numbers/jacobsthalnumbers.omdoc?jacobsthalnumbers?jacobsthalnumbershttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn

    In geschlossener Form haben wir

    (Jn)=(((2n)-((1)n))3)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/jacobsthalnumbers.omdoc?jacobsthalnumbers?jacobsthalnumbersnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn

  • Kanten en

    Ein gerichteter Graph (auch Digraph) ist ein Paar V,Ehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE aus einer Menge VV und einer Menge E(V,V)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetEhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairVV geordneter Paare über VV. Wir nennen VV die Knoten und EE die Kanten (auch Bögen) von GG.


  • Kanten en

    Ein Graph ist ein Paar V,Ehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE, so dass VV eine Menge ist und E(V,V)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetEhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairVV eine Teilmenge der Menge der Paare aus VV. Wir nennen VV die Ecken (oder Punkte, Knoten) and EE die Kanten (oder Linien, Pfeile) von GG.


  • Kantenanzahl Show Notations en

    Die Kantenanzahl (Anzahl der Kanten eines Graphen) GG bezeichnet man mit Ghttp://mathhub.info/smglom/graphs/graphnumberofedges.omdoc?graphnumberofedges?graphnumberofedgesG.


  • Kantengraph Show Notations en

    Für einen Graph G=(V,E)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalGhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE bezeichnen wir den Graph L(G)http://mathhub.info/smglom/graphs/graphlinegraph.omdoc?graphlinegraph?linegraphG auf EE, in dem x,yEhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetxyE genau dann als Ecken benachbart sind, wenn sie es als Kanten in GG sind, als den Kantengraph von GG.


  • Kardinalität Show Notations en

    Wir nennen eine Menge AA is endlich mit Kardinalität (#(A))http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityAhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, wenn es eine bijektive Funktion f:A({n|(n<(#(A)))})http://mathhub.info/smglom/sets/functions.omdoc?functions?funfAhttp://mathhub.info/smglom/sets/set.omdoc?set?rsetsthttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-numbernhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthannhttp://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityA gibt.

    Die Kardinalität einer Menge AA wird oft auch als #(A)http://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityA, #(A)http://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityA, #(A)http://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityA oder #(A)http://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityA geschrieben.


  • Kempner-Reihen Show Notations

    Kempner-Reihen Kmhttp://mathhub.info/smglom/smglom/kempnerseries.omdoc?kempnerseries?kempnerseriesm sind subharmonische Reihen, die aus der harmonischenReihe gebildet werden, indem alle Summanden weggelassen werden, deren Nenner im Zehnersystem die Ziffer mm (bzw. die Ziffernfolge der Zahl mm) enthalten:

    Kmhttp://mathhub.info/smglom/smglom/kempnerseries.omdoc?kempnerseries?kempnerseriesm

    Zum Beispiel

    (K9)=(1+(12)+(13)+(14)++(17)+(18)+(110)++(117)+(118)+(120)++(187)+(188)+(1100)+((1101)))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/smglom/kempnerseries.omdoc?kempnerseries?kempnerserieshttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?division

    (K9)=22.92067...http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?startswithhttp://mathhub.info/smglom/smglom/kempnerseries.omdoc?kempnerseries?kempnerseries

    (K314159)=2302582.3338637827...http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?startswithhttp://mathhub.info/smglom/smglom/kempnerseries.omdoc?kempnerseries?kempnerseries


  • kleinste gemeinsame Vielfache Show Notations en

    Das kleinste gemeinsame Vielfache (kgV) zweier oder mehrerer ganzer Zahlen ist die kleinste natürliche Zahl, die durch alle diese Zahlen teilbar ist.


  • kleinster Primfaktor Show Notations

    Ein Primfaktor einer natürlichen Zahl nn ist ein kleinster Primfaktor lpf(n)http://mathhub.info/smglom/smglom/leastprimefactor.omdoc?leastprimefactor?leastprimefactorn, wenn alle Primfaktoren von nn gleich oder größer als dieser sind.


  • Knoten en

    Ein gerichteter Graph (auch Digraph) ist ein Paar V,Ehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE aus einer Menge VV und einer Menge E(V,V)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetEhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairVV geordneter Paare über VV. Wir nennen VV die Knoten und EE die Kanten (auch Bögen) von GG.


  • Knoten en

    Ein Graph ist ein Paar V,Ehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE, so dass VV eine Menge ist und E(V,V)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetEhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairVV eine Teilmenge der Menge der Paare aus VV. Wir nennen VV die Ecken (oder Punkte, Knoten) and EE die Kanten (oder Linien, Pfeile) von GG.


  • Knödel-Zahl Show Notations en

    Eine Knödel-Zahl für eine gegebene positive ganze Zahl nn ist eine zusammengesetzte Zahl mm, für die für jedes zu mm teilerfremde imhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanim gilt

    (i(m-n))1modmhttp://mathhub.info/smglom/numberfields/congruence.omdoc?congruence?congruent-modulohttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationihttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionmnm

    Die Menge aller Knödel-Zahlen einer Zahl nn wird mit Knhttp://mathhub.info/smglom/numbers/knoedelnumber.omdoc?knoedelnumber?Knoedel-numbern bezeichnet.


  • Knödel-Zahlen Show Notations en

    Eine Knödel-Zahl für eine gegebene positive ganze Zahl nn ist eine zusammengesetzte Zahl mm, für die für jedes zu mm teilerfremde imhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanim gilt

    (i(m-n))1modmhttp://mathhub.info/smglom/numberfields/congruence.omdoc?congruence?congruent-modulohttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationihttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionmnm

    Die Menge aller Knödel-Zahlen einer Zahl nn wird mit Knhttp://mathhub.info/smglom/numbers/knoedelnumber.omdoc?knoedelnumber?Knoedel-numbern bezeichnet.


  • kommutativ en

    Ein Ring Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR heißt kommutativ, fallsRhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR ein kommutativer Magma ist.


  • Kommutator Show Notations en

    Ist Ghttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureG eine Gruppe und a,bGhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabG, so definieren wir den Kommutator von aa und bb als (fundefeq[ab][a,b])http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqabhttp://mathhub.info/smglom/algebra/commutator.omdoc?commutator?commutatorab. Er ist identisch mit der Einheit der Gruppe, genau dann wenn gg und hh kommutieren.


  • komplexen Zahlen b Show Notations en

    Die Menge http://mathhub.info/smglom/numberfields/complexnumbers.omdoc?complexnumbers?complex-number der komplexen Zahlen besteht aus Zahlen der Form a+(b𝑖)http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionahttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationbhttp://mathhub.info/smglom/numberfields/complexnumbers.omdoc?complexnumbers?imaginary-unit, mit a,bhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number. Wir nennen 𝑖http://mathhub.info/smglom/numberfields/complexnumbers.omdoc?complexnumbers?imaginary-unit imaginäre Einheit.


  • Komponenten en

    Eine Struktur fasst mehrere existierende mathematische Objekte (die Komponenten) zu einem neuen Objekt zusammen. Strukturen werden normalerweise als endliche Aufzählungen ihrer Komponenten gegeben, die durch spezielle Namen referenziert werden können.


  • konvergent en

    Wir nennen eine Folge konvergent, falls sie gegen einen Grenzwert gg konvergiert.


  • konvergiert en

    Ist M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd ein metrischer Raum und (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan eine Folge mit (ai)Mhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselaiM für alle ihttp://mathhub.info/smglom/sets/set.omdoc?set?insetihttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, so sagen wir daß (sequenceon[name.cvar.0]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonname.cvar.0n gegen gMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetgM konvergiert (wir nennen gg den Grenzwert oder limitLimes und schreiben limnanhttp://mathhub.info/smglom/calculus/sequenceConvergence.omdoc?sequenceConvergence?limitnOMFOREIGNscala.xml.Node$@6040b6ef ), falls es für jedes ϵ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanϵ ein n0http://mathhub.info/smglom/sets/set.omdoc?set?insetOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, so daß (d)<ϵhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthandϵ für alle nn0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methannOMFOREIGNscala.xml.Node$@6040b6ef.


  • konvergiert en

    Ist X,Ohttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureXOMFOREIGNscala.xml.Node$@6040b6ef ein Hausdorff-Raum, so sagen wir dass eine Folge (sequenceon[name.cvar.0]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonname.cvar.0n gegen xXhttp://mathhub.info/smglom/sets/set.omdoc?set?insetxX konvergiert (wir nennen xx den Hausdorff-Grenzwert und schreiben xx), falls jede Umgebung von xx nur endlich viele xiOMFOREIGNscala.xml.Node$@6040b6ef enthält.


  • konverse Relation Show Notations en

    (fundefeq[R]R-1)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqRhttp://mathhub.info/smglom/sets/converse-relation.omdoc?converse-relation?converse-relationR ist die konverse Relation von R(A×B)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetRhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAB.


  • konvex en

    Eine Menge S(n)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetShttp://mathhub.info/smglom/linear-algebra/nspace.omdoc?nspace?Euclidean-spacen heißt konvex, falls (bsetst[t](ta)+((1-t)b))Shttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsethttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstthttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationtahttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractiontbS für alle a,bShttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabS.


  • konvexe Hülle e en

    Die konvexe Hülle einer Menge S(n)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetShttp://mathhub.info/smglom/linear-algebra/nspace.omdoc?nspace?Euclidean-spacen ist der Schnitt aller konvexen Mengen, die SS enthalten.


  • konvexes Polytop en

    Ein konvexes Polytop ist definiert als die konvexe Hülle einer Menge von Punkten im Euklidischen Raum.


  • Kreuzprodukt Show Notations en

    Das Kreuzprodukt zweier Vektoren in 3http://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?nCartSpacehttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number ist ihr Vektorprodukt .


  • Kronecker-Delta Show Notations

    Das Kronecker-Delta ist definiert als:

    (δij)=(defined-piecewise(
    1wenn(i=j)
    )
    (
    0wenn(ij)
    )
    )
    http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/smglom/kroneckerdelta.omdoc?kroneckerdelta?kroneckerdeltaijhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalijhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?notequalij

    Dabei können ii und jj Elemente einer beliebigen Indexmenge II sein, meist jedoch einer endlichen Teilmenge der natürlichen Zahlen.


  • kubikfreie Taxicab-Zahl en

    Eine kubikfreie Taxicab-Zahl ist eine Taxicab-Zahl, die durch keine Kubikzahl außer 13http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiation teilbar ist.


  • kubischer Graph en

    Ein 3-regulärer Graph wird auch kubischer Graph genannt.


  • Kuchen-Zahl en

    Die Kuchen-Zahl Cnhttp://mathhub.info/smglom/numbers/cakenumber.omdoc?cakenumber?cakenumbern ist die grösste Zahl von Teilen, in die ein Würfel (oder ein 3-dimensionaler Raum) durch nn Ebenen geteilt werden kann.

    (multi-relation-expressionCnequal(𝒞3n)+(𝒞2n)+(𝒞1n)+(𝒞0n)equal(16)((n3)+(5n)+6))http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numbers/cakenumber.omdoc?cakenumber?cakenumbernhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficientnhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficientnhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficientnhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficientnhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationn

  • Kynea-Zahl en

    Eine Kynea-Zahl ist eine ganze Zahl der Form

    (((4n)+(2(n+1)))-1)=((((2n)+1)2)-2)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn

    mit einer positiven ganzen Zahl nn.


  • Körper en

    Ein Körper ist ein Ring in dem alle Elemente multiplikative Inverse haben.


  • Lagrange Notation en

    Ist f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN differenzierbar auf MM, so definieren wir die Ableitungsfunktion (Dxf):MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfxMN von ff nach xx als

    (fundefeq[fxa]Dxf(a))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfxahttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa

    Die Abhängigkeit von xx ist bleibt dieser Notation (Lagrange Notation) implizit. In Leibniz Notation schreiben wir die Ableitungsfunktion von ff nach xx als Dxfhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfx. In Eulers Notation schreiben wir Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa und in Newtonnotation Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa.


  • Laufindex en

    Wir definieren das Produkt über eine Folge aiOMFOREIGNscala.xml.Node$@6040b6ef durch

    (i=nmai):=(defined-piecewise(
    0wenn(n=m)
    )
    (
    (an)sonst
    )
    )
    http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdfromtonmiOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnmhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionOMFOREIGNscala.xml.Node$@6040b6ef

    Die Variable ii wird der Laufindex oder Multiplikationsindex genannt, nn die untere (oder Startwert) und mm die obereGrenze (oder Endwert) des Produkts, zusammen bestimmen sie den Multiplikationsbereich.

    Gebräuchlich sind auch die fogenden Varianten des Produktoperators: φaihttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdCollφOMFOREIGNscala.xml.Node$@6040b6ef und iSaihttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdInCollSiOMFOREIGNscala.xml.Node$@6040b6ef. Der erste spezifiziert den Multiplikationsbereich durch eine Formel φφ über ii und der zweite gibt den Multiplikationsbereich direkt als eine Menge SS an.


  • Laufindex en

    Summation ist iterierte Addition. Wir definieren die Summe über eine Folge aiOMFOREIGNscala.xml.Node$@6040b6ef durch

    (i=nmai):=(defined-piecewise(
    0wenn(n=m)
    )
    (
    (an)sonst
    )
    )
    http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/numberfields/sum.omdoc?sum?sumnmiOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnmhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionOMFOREIGNscala.xml.Node$@6040b6ef

    Die Variable ii wird der Laufindex oder Summationsindex genannt, nn die untere (oder Startwert) und mm die obereGrenze (oder Endwert) der Summe, zusammen bestimmen sie den Summationsbereich.

    Gebräuchlich sind auch die fogenden Varianten des Summenoperators: (SumCollφ[i]ai)http://mathhub.info/smglom/numberfields/sum.omdoc?sum?SumCollφiOMFOREIGNscala.xml.Node$@6040b6ef und iSaihttp://mathhub.info/smglom/numberfields/sum.omdoc?sum?SumInCollSiOMFOREIGNscala.xml.Node$@6040b6ef. Der erste spezifiziert den Summationsbereich durch eine Formel φφ über ii und der zweite gibt den Summationsbereich als eine Menge SS an.


  • Lebesgue-Identität en

    Für alle Zahlen mm, nn, pp und qq gilt die Lebesgue-Identität:

    (((m2)+(n2)+(p2)+(q2))2)=((((2mq)+(2np))2)+(((2nq)-(2mp))2)+((((m2)+(n2))-(p2)-(q2))2))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationqhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationmqhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnqhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationmphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationq

  • leere Menge Show Notations en tr ro

    Die leere Menge http://mathhub.info/smglom/sets/emptyset.omdoc?emptyset?eset (schreibe auch http://mathhub.info/smglom/sets/emptyset.omdoc?emptyset?eset) ist die Menge ohne Elemente.


  • leere Produkt Show Notations en

    Das leere Produkt ist in der Mathematik der Sonderfall eines Produktes mit null Faktoren. Ihm wird der Wert 1 zugewiesen.


  • leere Summe Show Notations en

    Die leere Summe ist in der Mathematik der Sonderfall einer Summe mit onull Summanden. Der leeren Summe wird der Wert 0, das neutrale Element der Addition, zugewiesen.

    Zum Beispiel gilt für m>nhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanmn

    i=mnaihttp://mathhub.info/smglom/numberfields/sum.omdoc?sum?summniOMFOREIGNscala.xml.Node$@6040b6ef

  • leeren Summe Show Notations en

    Die leere Summe ist in der Mathematik der Sonderfall einer Summe mit onull Summanden. Der leeren Summe wird der Wert 0, das neutrale Element der Addition, zugewiesen.

    Zum Beispiel gilt für m>nhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanmn

    i=mnaihttp://mathhub.info/smglom/numberfields/sum.omdoc?sum?summniOMFOREIGNscala.xml.Node$@6040b6ef

  • Leibniz Notation en

    Ist f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN differenzierbar auf MM, so definieren wir die Ableitungsfunktion (Dxf):MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfxMN von ff nach xx als

    (fundefeq[fxa]Dxf(a))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfxahttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa

    Die Abhängigkeit von xx ist bleibt dieser Notation (Lagrange Notation) implizit. In Leibniz Notation schreiben wir die Ableitungsfunktion von ff nach xx als Dxfhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfx. In Eulers Notation schreiben wir Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa und in Newtonnotation Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa.


  • Leyland-Zahl en

    Eine Leyland-Zahl ist eine ganze Zahl der Form

    (xy)+(yx)http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationxyhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationyx

    Dabei sind xx und yy ganze Zahlen größer als 1.

    Die ersten Leyland-Zahlen sind 8,17,32,54,57,100,145,177,320,368,512,...http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?infseq


  • Limes Show Notations en

    Seien A,dAhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureAOMFOREIGNscala.xml.Node$@6040b6ef und B,dBhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureBOMFOREIGNscala.xml.Node$@6040b6ef metrische Räume und f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN für MAhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetMA und NBhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetNB, dann nennen wir lMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetlM en Grenzwert (oder Limes) von ff wenn xx gegen einen Häufungspunkt pNhttp://mathhub.info/smglom/sets/set.omdoc?set?insetpN strebt (schreibe limxp(f)http://mathhub.info/smglom/calculus/functionlimit.omdoc?functionlimit?limitpxf), falls es für jedes ϵ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanϵ ein δ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanδ gibt, so dass für alle xx mit (multi-relation-expression0lethan(dB)lethanδ)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanδ auch (dA)ϵhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanOMFOREIGNscala.xml.Node$@6040b6efϵ gilt.


  • limit Show Notations

    Ist M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd ein metrischer Raum und (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan eine Folge mit (ai)Mhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselaiM für alle ihttp://mathhub.info/smglom/sets/set.omdoc?set?insetihttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, so sagen wir daß (sequenceon[name.cvar.0]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonname.cvar.0n gegen gMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetgM konvergiert (wir nennen gg den Grenzwert oder limitLimes und schreiben limnanhttp://mathhub.info/smglom/calculus/sequenceConvergence.omdoc?sequenceConvergence?limitnOMFOREIGNscala.xml.Node$@6040b6ef ), falls es für jedes ϵ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanϵ ein n0http://mathhub.info/smglom/sets/set.omdoc?set?insetOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, so daß (d)<ϵhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthandϵ für alle nn0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methannOMFOREIGNscala.xml.Node$@6040b6ef.


  • lineare Ordnung en

    Eine partielle Ordnung RR heißt totale Ordnung (or einfache Ordnung oder lineare Ordnung), falls abhttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?poleab oder bahttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?poleba für alle a,bAhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabA.

    Wir nennen eine Struktur S,polehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureShttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole aus einer Menge SS und einer totalen Ordnung polehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole eine total geordnete Menge.


  • Linien en

    Ein Graph ist ein Paar V,Ehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE, so dass VV eine Menge ist und E(V,V)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetEhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairVV eine Teilmenge der Menge der Paare aus VV. Wir nennen VV die Ecken (oder Punkte, Knoten) and EE die Kanten (oder Linien, Pfeile) von GG.


  • link teilbar en

    Sei Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR ein Ring und a,bRhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabR mit a0http://mathhub.info/smglom/mv/equal.omdoc?equal?notequala, dann nennen wir aa einen linken Teiler von bb (und sagen aa teilt bb links oder bb ist link teilbar durch aa) und bb ein linkes Vielfaches von aa, wenn es ein nRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnR gibt, für die (multiplication)=bhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationb ist.

    Analog nennen wir aa einen rechten Teiler von bb (und sagen aa teilt bb rechts oder bb ist rechts teilbar durch aa) und bb ein rechtes Vielfaches von aa, wenn es ein nRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnR gibt, für die (na)bhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnab ist.

    Ist RR kommutativ, so fallen rechte und linke Teiler zusammen, und wir sprechen einfach von Teiler, teilbar, und Vielfachen. Wir schreiben dann a|bhttp://mathhub.info/smglom/algebra/ring-divisor.omdoc?ring-divisor?divisorab.

    Wir nennen 1, 1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction, nn und nhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn triviale divisoren von nn. Ein Teiler von nn, der nicht trivial ist, heißt echter Teiler von nn.


  • linken Teiler en

    Sei Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR ein Ring und a,bRhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabR mit a0http://mathhub.info/smglom/mv/equal.omdoc?equal?notequala, dann nennen wir aa einen linken Teiler von bb (und sagen aa teilt bb links oder bb ist link teilbar durch aa) und bb ein linkes Vielfaches von aa, wenn es ein nRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnR gibt, für die (multiplication)=bhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationb ist.

    Analog nennen wir aa einen rechten Teiler von bb (und sagen aa teilt bb rechts oder bb ist rechts teilbar durch aa) und bb ein rechtes Vielfaches von aa, wenn es ein nRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnR gibt, für die (na)bhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnab ist.

    Ist RR kommutativ, so fallen rechte und linke Teiler zusammen, und wir sprechen einfach von Teiler, teilbar, und Vielfachen. Wir schreiben dann a|bhttp://mathhub.info/smglom/algebra/ring-divisor.omdoc?ring-divisor?divisorab.

    Wir nennen 1, 1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction, nn und nhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn triviale divisoren von nn. Ein Teiler von nn, der nicht trivial ist, heißt echter Teiler von nn.


  • linkes Vielfaches en

    Sei Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR ein Ring und a,bRhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabR mit a0http://mathhub.info/smglom/mv/equal.omdoc?equal?notequala, dann nennen wir aa einen linken Teiler von bb (und sagen aa teilt bb links oder bb ist link teilbar durch aa) und bb ein linkes Vielfaches von aa, wenn es ein nRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnR gibt, für die (multiplication)=bhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationb ist.

    Analog nennen wir aa einen rechten Teiler von bb (und sagen aa teilt bb rechts oder bb ist rechts teilbar durch aa) und bb ein rechtes Vielfaches von aa, wenn es ein nRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnR gibt, für die (na)bhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnab ist.

    Ist RR kommutativ, so fallen rechte und linke Teiler zusammen, und wir sprechen einfach von Teiler, teilbar, und Vielfachen. Wir schreiben dann a|bhttp://mathhub.info/smglom/algebra/ring-divisor.omdoc?ring-divisor?divisorab.

    Wir nennen 1, 1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction, nn und nhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn triviale divisoren von nn. Ein Teiler von nn, der nicht trivial ist, heißt echter Teiler von nn.


  • linksseitige Grenzwert Show Notations en

    Sei f:SThttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfST gegeben mit S,Thttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?msseteqSThttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number und ahttp://mathhub.info/smglom/sets/set.omdoc?set?insetahttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number.

    Dann ist der linksseitige Grenzwert von ff in aa (auch: der Grenzwert von ff wenn xx von unten gegen pp strebt) definiert als diejenige Zahl lhttp://mathhub.info/smglom/sets/set.omdoc?set?insetlhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number, so dass für jedes ϵ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanϵ ein δ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanδ gibt so dass (|(((fx))l)|)ϵhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/numberfields/absolutevalue.omdoc?absolutevalue?absolute-valuehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationfxlϵ für alle (multi-relation-expression0lethan(a)xlethanδ)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionaxhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanδ. Der linksseitige Grenzwert wird geschrieben als limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?leftsided-limitaxf, limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?leftsided-limitaxf oder limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?leftsided-limitaxf.

    Analog definieren wir den rechtsseitigen Grenzwert von ff in aa (auch: der Grenzwert von ff wenn xx von unten gegen pp strebt) definiert als diejenige Zahl lhttp://mathhub.info/smglom/sets/set.omdoc?set?insetlhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number, so dass für jedes ϵ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanϵ ein δ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanδ gibt so dass (|(((f))l)|)ϵhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/numberfields/absolutevalue.omdoc?absolutevalue?absolute-valuehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionflϵ für alle (multi-relation-expression0lethan(x)alethanδ)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionxahttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanδ. Der linksseitige Grenzwert wird geschrieben als limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?rightsided-limitaxf, limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?rightsided-limitaxf oder limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?rightsided-limitaxf.


  • linksseitige Grenzwert Show Notations en

    Sei f:SThttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfST gegeben mit S,Thttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?msseteqSThttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number und ahttp://mathhub.info/smglom/sets/set.omdoc?set?insetahttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number.

    Dann ist der linksseitige Grenzwert von ff in aa (auch: der Grenzwert von ff wenn xx von unten gegen pp strebt) definiert als diejenige Zahl lhttp://mathhub.info/smglom/sets/set.omdoc?set?insetlhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number, so dass für jedes ϵ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanϵ ein δ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanδ gibt so dass (|(((fx))l)|)ϵhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/numberfields/absolutevalue.omdoc?absolutevalue?absolute-valuehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationfxlϵ für alle (multi-relation-expression0lethan(a)xlethanδ)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionaxhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanδ. Der linksseitige Grenzwert wird geschrieben als limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?leftsided-limitaxf, limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?leftsided-limitaxf oder limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?leftsided-limitaxf.

    Analog definieren wir den rechtsseitigen Grenzwert von ff in aa (auch: der Grenzwert von ff wenn xx von unten gegen pp strebt) definiert als diejenige Zahl lhttp://mathhub.info/smglom/sets/set.omdoc?set?insetlhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number, so dass für jedes ϵ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanϵ ein δ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanδ gibt so dass (|(((f))l)|)ϵhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/numberfields/absolutevalue.omdoc?absolutevalue?absolute-valuehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionflϵ für alle (multi-relation-expression0lethan(x)alethanδ)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionxahttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanδ. Der linksseitige Grenzwert wird geschrieben als limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?rightsided-limitaxf, limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?rightsided-limitaxf oder limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?rightsided-limitaxf.


  • Linkstrunkierbare Primzahlen en

    Linkstrunkierbare Primzahlen sind Primzahlen, in denen an keiner Stelle die Ziffer Null steht und bei denen das Weglassen einer beliebigen Anzahl führender Stellen wieder zu einer Primzahl führt.

    Zum Beispiel: 937

    Beidseitig trunkierbare Primzahlen sind sowohl links- als auch rechtstrunkierbare Primzahlen.

    Zum Beispiel: 3137


  • Liouville-Konstante en

    Die Liouville-Konstante ist eine Liouvillesche Zahl, definiert durch

    (infinite-sum1[j]10((j!)))http://mathhub.info/smglom/calculus/infinitesum.omdoc?infinitesum?infinite-sumjhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/factorial.omdoc?factorial?factorialj

  • Liouvillesche Zahl en

    Eine Liouvillesche Zahl ist eine reelle Zahl xx, welche die Bedingung erfüllt, dass für alle positiven ganzen Zahlen nn ganze Zahlen pp und qq mit q>1http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanq existieren, so dass

    (multi-relation-expression0lethan|(x-(pq))|lethan1(qn))http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/numberfields/absolutevalue.omdoc?absolutevalue?absolute-valuehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionpqhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationqn

  • Lobb-Zahl Show Notations en

    Eine Lobb-Zahl Lm,nhttp://mathhub.info/smglom/numbers/lobbnumbers.omdoc?lobbnumbers?Lobb-numbermn ist die Anzahl der Möglichkeiten n+mhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnm öffnende Klammern und n-mhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnm schließende Klammern so anzuordnen, dass sie den Beginn einer korrekten Folge von öffnenden und schließenden Klammern bilden.

    (Lm,n)=((((2m)+1)(m+n+1))(𝒞(m+n)(2n)))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/lobbnumbers.omdoc?lobbnumbers?Lobb-numbermnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionmnhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficienthttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionmn

    Dabei sind mm und nn zwei ganze Zahlen mit (multi-relation-expressionnmethanmmethan0)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionnhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methanmhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methan.


  • Lucas-Polynome Show Notations en

    Die Lucas-Polynome werden rekursiv definiert

    ((Ln(x))x)=(defined-piecewise(
    2wenn(n=0)
    )
    (
    xwenn(n=1)
    )
    (
    ((x(L(n-1)(x))x)+((L(n-2)(x))x))wenn(n2)
    )
    )
    http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/lucaspolynomials.omdoc?lucaspolynomials?Lucas-polynomialsnxhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecexhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationxhttp://mathhub.info/smglom/smglom/lucaspolynomials.omdoc?lucaspolynomials?Lucas-polynomialshttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/lucaspolynomials.omdoc?lucaspolynomials?Lucas-polynomialshttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnxhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methann

    Die ersten Lucas-Polynome sind:

    ((L0(x))x)=2http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/lucaspolynomials.omdoc?lucaspolynomials?Lucas-polynomialsx

    ((L1(x))x)=xhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/lucaspolynomials.omdoc?lucaspolynomials?Lucas-polynomialsxx

    ((L2(x))x)=((x2)+2)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/lucaspolynomials.omdoc?lucaspolynomials?Lucas-polynomialsxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationx

    ((L3(x))x)=((x3)+(3x))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/lucaspolynomials.omdoc?lucaspolynomials?Lucas-polynomialsxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationx

    ((L4(x))x)=((x4)+(4(x2))+2)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/lucaspolynomials.omdoc?lucaspolynomials?Lucas-polynomialsxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationx

    ((L5(x))x)=((x5)+(5(x3))+(5x))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/smglom/lucaspolynomials.omdoc?lucaspolynomials?Lucas-polynomialsxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationx


  • Lucas-Zahlen Show Notations en

    Die Lucas-Zahlen werden rekursiv definiert:

    L0http://mathhub.info/smglom/numbers/lucasnumbers.omdoc?lucasnumbers?Lucas-numbers equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal 2
    L1http://mathhub.info/smglom/numbers/lucasnumbers.omdoc?lucasnumbers?Lucas-numbers equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal 1
    Lnhttp://mathhub.info/smglom/numbers/lucasnumbers.omdoc?lucasnumbers?Lucas-numbersn equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal (L(n-1))+(L(n-2))http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numbers/lucasnumbers.omdoc?lucasnumbers?Lucas-numbershttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numbers/lucasnumbers.omdoc?lucasnumbers?Lucas-numbershttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn

  • Lychrel-Zahl en

    Eine Lychrel-Zahl ist eine natürliche Zahl, die niemals palindromisch wird, wenn man wiederholt die Zahl addiert, die durch Umkehrung der Ziffernfolge entsteht.


  • Länge en

    Ist G=(V,E)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalGhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE gerichteter Graph, so nennen wir einen Vektor (p=((v0,,vn)))(V(n+1))http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalphttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlivnhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?nCartSpaceVhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionn einen Pfad in GG wenn (vi-1,vi)Ehttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6efE für alle 1inhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?betweeneein mit n>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethann.

    • v0OMFOREIGNscala.xml.Node$@6040b6ef heißt der Startknoten von pp (schreibe start(p)http://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pstartp )

    • vnOMFOREIGNscala.xml.Node$@6040b6ef heißt der Endknoten von pp (schreibe end(p)http://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pendp )

    • nn heißt die Länge von pp (schreibe len(p)http://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?plenp )


  • mal differeinzierbar en

    Wir nennen ff nn mal differeinzierbar in aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM, falls dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa als Grenzwert existiert. Analog nennen wir ff nn mal differenzierbar auf MM, falls dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx existiert und auf MM total ist.

    Wir nennen ff unendlich differenzierbar oder glatt auf aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM oder MM, wenn dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa beziehungsweise dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx für alle nhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number existieren.


  • mal differenzierbar en

    Wir nennen ff nn mal differeinzierbar in aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM, falls dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa als Grenzwert existiert. Analog nennen wir ff nn mal differenzierbar auf MM, falls dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx existiert und auf MM total ist.

    Wir nennen ff unendlich differenzierbar oder glatt auf aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM oder MM, wenn dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa beziehungsweise dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx für alle nhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number existieren.


  • maximale Läenge en

    Für einen Graph GG ist die minimale Länge eines in GG enthaltenen Kreises die Taillenweite g(G)http://mathhub.info/smglom/graphs/graphcycleparameters.omdoc?graphcycleparameters?girthG von GG, die maximale Läenge eines in GG enthaltenen Kreises ist der Umfang von GG. Für einen Graph, der keinen Kreis enthält, setzen wir die Taillenweite auf (g(G))=http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/graphs/graphcycleparameters.omdoc?graphcycleparameters?girthGhttp://mathhub.info/smglom/numberfields/infinity.omdoc?infinity?infinity, sein Umfang wird auf Null gesetzt.


  • Maximum Show Notations en

    Das Minimum (minimumS)http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?minimumS (Maximum (maximumS)http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?maximumS) einer geordneten Menge SS ist dasjenige Element mm (wenn es existiert), so dass alle anderen Elemente von SS größer (kleiner) sind.

    Ist ee ein Ausdruck und φφ eine Bedingung (in einer Variablen xx), so schreiben wir (bmaxvalφ[x]e)http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?bmaxvalφxe für (maximum(bsetst[x]φ))http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?maximumhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstxφ und nennen es das maximum für ee über φφ. Analog schreiben wir (bminvalφ[x]e)http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?bminvalφxe für (minimum(bsetst[x]φ))http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?minimumhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstxφ und nennen es das minimum für ee über φφ.


  • Mclaurinreihe

    Sei ff eine reellwertige oder komplexe Funktion die glatt ist auf einem Häufungspunkt aa des Definitionsbereichs von ff, dann nennen wir die Reihe

    (fundefeq[fxa]𝑇f(x;a))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfxahttp://mathhub.info/smglom/calculus/Taylor-series.omdoc?Taylor-series?Taylor-seriesfxa

    die Taylorreihe für ff am Entwicklungspunkt aa. Ist a=0http://mathhub.info/smglom/mv/equal.omdoc?equal?equala, so nennen wir die Reihe die Mclaurinreihe.


  • Meertens-Zahl en

    Eine Meertens-Zahl ist eine ganze Zahl, die gleich ihrer Gödelnummer ist. Die einzige bekannte Meertens-Zahl ist ‘

    81312000=((28)(31)(53)(71)(112)(130)(170)(190))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiation

  • Mercator-Reihe en

    Die Mercator-Reihe oder Newton-Mercator-Reihe ist die Taylorreihe des natürlichen Logarithmus.

    ln((1+x))http://mathhub.info/smglom/calculus/naturallogarithm.omdoc?naturallogarithm?natural-logarithmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionx

    Die Reihe konvergiert für (multi-relation-expression1lethanxlessthan1)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanxhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthan.


  • Mersenne-Primzahl Show Notations en

    Eine Mersenne-Primzahl ist eine Primzahl der Form

    (Mn)=((2n)-1)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/primes/mersenneprime.omdoc?mersenneprime?Mersenne-primenhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn

    Dann ist nn auch eine Primzahl.


  • Mersenne-Zahl Show Notations en

    Eine Mersenne-Zahl ist eine natürliche Zahl der Form (Mn)=((2n)-1)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/mersennenumber.omdoc?mersennenumber?Mersenne-numbernhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn


  • Mertens-Function

    Die Mertens-Function ist für alle positiven integers nn definiert:

    MM

    wobei μkhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationμk die Möbius-Funktion ist.

    Die Funktion ist nach Franz Mertens benannt.

    Die Definition kann auf die reellen Zahlen erweitert werden:

    MM

  • Metrik

    Für eine Mente MM nennen wir eine Funktion d:(M×M)http://mathhub.info/smglom/sets/functions.omdoc?functions?fundhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsMMhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number eine Abstandsfunktion (or Metrik ) auf MM, falls die folgenden drei Eigenschaften für alle x,y,zMhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetxyzM gelten:

    1. 1.

      (d)=0http://mathhub.info/smglom/mv/equal.omdoc?equal?equald gdw. x=yhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalxy (Definitheit),

    2. 2.

      (d)=(d)http://mathhub.info/smglom/mv/equal.omdoc?equal?equaldd (Symmetrie) und

    3. 3.

      (d)((d)+(d))http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethandhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additiondd (Dreiecksungleichung).

    Wir nennen M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd einen metrischen Raum mit Grundmenge MM und Metrik dd.


  • Metrik ) en

    Für eine Mente MM nennen wir eine Funktion d:(M×M)http://mathhub.info/smglom/sets/functions.omdoc?functions?fundhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsMMhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number eine Abstandsfunktion (or Metrik ) auf MM, falls die folgenden drei Eigenschaften für alle x,y,zMhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetxyzM gelten:

    1. 1.

      (d)=0http://mathhub.info/smglom/mv/equal.omdoc?equal?equald gdw. x=yhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalxy (Definitheit),

    2. 2.

      (d)=(d)http://mathhub.info/smglom/mv/equal.omdoc?equal?equaldd (Symmetrie) und

    3. 3.

      (d)((d)+(d))http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethandhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additiondd (Dreiecksungleichung).

    Wir nennen M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd einen metrischen Raum mit Grundmenge MM und Metrik dd.


  • metrischen Raum en

    Für eine Mente MM nennen wir eine Funktion d:(M×M)http://mathhub.info/smglom/sets/functions.omdoc?functions?fundhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsMMhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number eine Abstandsfunktion (or Metrik ) auf MM, falls die folgenden drei Eigenschaften für alle x,y,zMhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetxyzM gelten:

    1. 1.

      (d)=0http://mathhub.info/smglom/mv/equal.omdoc?equal?equald gdw. x=yhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalxy (Definitheit),

    2. 2.

      (d)=(d)http://mathhub.info/smglom/mv/equal.omdoc?equal?equaldd (Symmetrie) und

    3. 3.

      (d)((d)+(d))http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethandhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additiondd (Dreiecksungleichung).

    Wir nennen M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd einen metrischen Raum mit Grundmenge MM und Metrik dd.


  • minimale Länge en

    Für einen Graph GG ist die minimale Länge eines in GG enthaltenen Kreises die Taillenweite g(G)http://mathhub.info/smglom/graphs/graphcycleparameters.omdoc?graphcycleparameters?girthG von GG, die maximale Läenge eines in GG enthaltenen Kreises ist der Umfang von GG. Für einen Graph, der keinen Kreis enthält, setzen wir die Taillenweite auf (g(G))=http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/graphs/graphcycleparameters.omdoc?graphcycleparameters?girthGhttp://mathhub.info/smglom/numberfields/infinity.omdoc?infinity?infinity, sein Umfang wird auf Null gesetzt.


  • Minimum Show Notations en

    Das Minimum (minimumS)http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?minimumS (Maximum (maximumS)http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?maximumS) einer geordneten Menge SS ist dasjenige Element mm (wenn es existiert), so dass alle anderen Elemente von SS größer (kleiner) sind.

    Ist ee ein Ausdruck und φφ eine Bedingung (in einer Variablen xx), so schreiben wir (bmaxvalφ[x]e)http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?bmaxvalφxe für (maximum(bsetst[x]φ))http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?maximumhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstxφ und nennen es das maximum für ee über φφ. Analog schreiben wir (bminvalφ[x]e)http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?bminvalφxe für (minimum(bsetst[x]φ))http://mathhub.info/smglom/numberfields/minmax.omdoc?minmax?minimumhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstxφ und nennen es das minimum für ee über φφ.


  • Mirpzahlen en

    Mirpzahlen („prim“ rückwärts geschrieben) sind Primzahlen, die rückwärts gelesen eine andere Primzahl ergeben.


  • Motzkin Zahlen Show Notations en

    Motzkin Zahlen Mnhttp://mathhub.info/smglom/numbers/motzkinnumberrec.omdoc?motzkinnumberrec?Motzkin-numbern werden rekursiv definiert:

    M(n+1)http://mathhub.info/smglom/numbers/motzkinnumberrec.omdoc?motzkinnumberrec?Motzkin-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionn

    Dabei ist (multi-relation-expressionM0equalM1equal1)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numbers/motzkinnumberrec.omdoc?motzkinnumberrec?Motzkin-numberhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/motzkinnumberrec.omdoc?motzkinnumberrec?Motzkin-numberhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal.


  • Motzkin-Primzahl en

    Eine Motzkin-Primzahl ist eine Motzkin-Zahl, die auch Primzahl ist.


  • Motzkin-Zahl Show Notations en

    Die Motzkin-Zahl Mnhttp://mathhub.info/smglom/numbers/motzkinnumber.omdoc?motzkinnumber?Motzkin-numbern zu einer gegebenen natürlichen Zahl nn ist die Anzahl der unterschiedlichen Möglichkeiten in einem Kreis zwischen nn Punkten sich nicht schneidende Sehnen zu zeichnen.

    Die ersten Motzkin-Zahlen sind 1,1,2,4,9,21,51,127,323,835,2188,5798,...http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?infseq


  • Motzkin-Zahlen

    Motzkin-Zahlen Mnhttp://mathhub.info/smglom/numbers/motzkinnumbercat.omdoc?motzkinnumbercat?motzkinnumbercatn für natürliche Zahl nn können durch Catalan-Zahlen Ckhttp://mathhub.info/smglom/numbers/catalannumber.omdoc?catalannumber?catalannumberk ausgedrückt werden:

    Mnhttp://mathhub.info/smglom/numbers/motzkinnumbercat.omdoc?motzkinnumbercat?motzkinnumbercatn

  • multiperfekte Zahl en

  • Multiplikation Show Notations

    Die arithmetischen Operationen sind Addition additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?addition, Subtraktion, subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction, Multiplikation multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplication, Division divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?division und Exponentiation.


  • Multiplikationsbereich en

    Wir definieren das Produkt über eine Folge aiOMFOREIGNscala.xml.Node$@6040b6ef durch

    (i=nmai):=(defined-piecewise(
    0wenn(n=m)
    )
    (
    (an)sonst
    )
    )
    http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdfromtonmiOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnmhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionOMFOREIGNscala.xml.Node$@6040b6ef

    Die Variable ii wird der Laufindex oder Multiplikationsindex genannt, nn die untere (oder Startwert) und mm die obereGrenze (oder Endwert) des Produkts, zusammen bestimmen sie den Multiplikationsbereich.

    Gebräuchlich sind auch die fogenden Varianten des Produktoperators: φaihttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdCollφOMFOREIGNscala.xml.Node$@6040b6ef und iSaihttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdInCollSiOMFOREIGNscala.xml.Node$@6040b6ef. Der erste spezifiziert den Multiplikationsbereich durch eine Formel φφ über ii und der zweite gibt den Multiplikationsbereich direkt als eine Menge SS an.


  • Multiplikationsindex en

    Wir definieren das Produkt über eine Folge aiOMFOREIGNscala.xml.Node$@6040b6ef durch

    (i=nmai):=(defined-piecewise(
    0wenn(n=m)
    )
    (
    (an)sonst
    )
    )
    http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdfromtonmiOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnmhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionOMFOREIGNscala.xml.Node$@6040b6ef

    Die Variable ii wird der Laufindex oder Multiplikationsindex genannt, nn die untere (oder Startwert) und mm die obereGrenze (oder Endwert) des Produkts, zusammen bestimmen sie den Multiplikationsbereich.

    Gebräuchlich sind auch die fogenden Varianten des Produktoperators: φaihttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdCollφOMFOREIGNscala.xml.Node$@6040b6ef und iSaihttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdInCollSiOMFOREIGNscala.xml.Node$@6040b6ef. Der erste spezifiziert den Multiplikationsbereich durch eine Formel φφ über ii und der zweite gibt den Multiplikationsbereich direkt als eine Menge SS an.


  • multiplikative Struktur en

    Eine Struktur Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR heißt Ring, wenn Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR eine Abelsch e Gruppe ist (die additive Struktur), Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR ein Monoid (die multiplikative Struktur) ist, sowie Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR is a Ringoid.

    Wir nennen 0 die Null des Rings und entsprechend 1 die Eins.


  • multirelationionaler Ausdruck Show Notations en

    Ein multirelationionaler Ausdruck steht für eine Konjunktion von relationalen Aussagen: (multi-relation-expressionaRbSc)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionaRbShttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationc gilt, falls (R)R gilt und ausserdem (multi-relation-expressionbSc)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionbSc.


  • Nachbar en

    Zwei Ecken xx und yy in einem Graph GG sind adjazent, oder Nachbarn, wenn x,yhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairxy eine Kante von GG ist.

    Zwei Kanten efhttp://mathhub.info/smglom/mv/equal.omdoc?equal?notequalef sind adjazent, wenn sie ein Ende gemeinsam haben.


  • Nah-Wilson-Primzahl en

    Eine Nah-Wilson-Primzahl ist eine Primzahl pp, für die folgende Kongruenz gilt:

    ((p-1)!)((1)+(Bp))mod(p2)http://mathhub.info/smglom/numberfields/congruence.omdoc?congruence?congruent-modulohttp://mathhub.info/smglom/numberfields/factorial.omdoc?factorial?factorialhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationBphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationp

    Hierbei ist B0http://mathhub.info/smglom/mv/equal.omdoc?equal?notequalB eine ganze Zahl mit kleinem Absolutwert |B|http://mathhub.info/smglom/numberfields/absolutevalue.omdoc?absolutevalue?absolute-valueB.


  • Narayana-Zahlen Show Notations en

    Die Narayana-Zahlen N(n,k)http://mathhub.info/smglom/numbers/narayananumber.omdoc?narayananumber?Narayana-numbernk sind die Anzahl der Wege von 0,0http://mathhub.info/smglom/sets/pair.omdoc?pair?pair nach (2n),0http://mathhub.info/smglom/sets/pair.omdoc?pair?pairhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationn unter Verwendung von Schritten nach Nordost und Südost ohne dabei unter die xx-Achse zu gelangen und mit genau kk Spitzen(Maxima).

    (N(n,k))=((1n)(𝒞kn)(𝒞(k-1)n))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/narayananumber.omdoc?narayananumber?Narayana-numbernkhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionnhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficientnkhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficientnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionk

    Dabei sind nn und kk natürliche Zahlen mit k1nhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?betweeneekn.


  • natürliche Dichte en

    Eine Teilmenge AA der positiven ganzen Zahlen hat eine asymptotische Dichte (oder natürliche Dichte) d(A)http://mathhub.info/smglom/smglom/asymptoticdensity.omdoc?asymptoticdensity?asymptotic-densityA, wobei (multi-relation-expression0lessthand(A)lessthan1)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthanhttp://mathhub.info/smglom/smglom/asymptoticdensity.omdoc?asymptoticdensity?asymptotic-densityAhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthan gilt, wenn der Grenzwert existiert

    d(A)http://mathhub.info/smglom/smglom/asymptoticdensity.omdoc?asymptoticdensity?asymptotic-densityA

    anhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationan ist die Anzahl der Elemente von AA, die nicht größer als nn sind.


  • natürliche Logarithmus Show Notations en

    Der natürliche Logarithmus ist der Logarithmus logarithm zur Basis ehttp://mathhub.info/smglom/numberfields/eulersnumber.omdoc?eulersnumber?eulersnumber.


  • natürlichen Zahlen Show Notations en

    Die Menge http://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number der natürlichen Zahlen ist die Menge {(0,1,2,)}http://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?aseqdots.

    Die Menge http://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?positive-natural-number der positiven natürlichen Zahlen ist {(1,2,3,)}http://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?aseqdots.


  • negativen ganzen Zahlen Show Notations en

    Die Menge der negativen ganzen Zahlen http://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?negative-integers ist {(,(3),(2),(1))}http://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?dotsaseqhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction.


  • negativen ganzen Zahlen Show Notations en

    Die Menge der negativen ganzen Zahlen http://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?negative-integers ist {(,(3),(2),(1))}http://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?dotsaseqhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction.


  • negativen rationalen Zahlen Show Notations

    Die Menge http://mathhub.info/smglom/numberfields/rationalnumbers.omdoc?rationalnumbers?rational-numbers der rationalen Zahlen ist die Menge (bsetst[nm]nm)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstnmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionnm.

    Wir schreiben http://mathhub.info/smglom/numberfields/rationalnumbers.omdoc?rationalnumbers?negative-rational-numbers und http://mathhub.info/smglom/numberfields/rationalnumbers.omdoc?rationalnumbers?positive-rational-numbers für die Mengen der negativen rationalen Zahlen und der positiven rationalen Zahlen.


  • negativen rationalen Zahlen Show Notations

    Die Menge http://mathhub.info/smglom/numberfields/rationalnumbers.omdoc?rationalnumbers?rational-numbers der rationalen Zahlen ist die Menge (bsetst[nm]nm)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstnmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionnm.

    Wir schreiben http://mathhub.info/smglom/numberfields/rationalnumbers.omdoc?rationalnumbers?negative-rational-numbers und http://mathhub.info/smglom/numberfields/rationalnumbers.omdoc?rationalnumbers?positive-rational-numbers für die Mengen der negativen rationalen Zahlen und der positiven rationalen Zahlen.


  • negativen reellen Zahlen Show Notations en

    Die Menge http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number der reellen Zahlen reelleZahlen ist die Vervollständigung von http://mathhub.info/smglom/numberfields/rationalnumbers.omdoc?rationalnumbers?rational-numbers.

    Wir schreiben http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?negative-real-number und http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?positive-real-number für die Mengen der negativen reellen Zahlen und der positiven reellen Zahlen.


  • neunte Smarandache-Konstante

    Die neunte Smarandache-Konstante http://mathhub.info/smglom/smglom/smarandacheconstant9.omdoc?smarandacheconstant9?ninth-smarandache-constant ist definiert als

    http://mathhub.info/smglom/smglom/smarandacheconstant9.omdoc?smarandacheconstant9?ninth-smarandache-constant

    Dabei ist S(n)http://mathhub.info/smglom/smglom/smarandachefunction.omdoc?smarandachefunction?smarandache-functionn die Smarandache-Funktion.

    Die Summe konvergiert.


  • Newman-Shanks-Williams-Primzahl en

    Eine Newman-Shanks-Williams-Primzahl, oder NSW-Primzahl , ist eine Primzahl pp, die in folgender Form geschrieben werden kann

    p=((((1+(2))((2m)+1))+((1-(2))((2m)+1)))2)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?square-roothttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?square-roothttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationm

    Dabei ist mm eine natürliche Zahl.


  • Newton en

    Ist f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN differenzierbar auf MM, so definieren wir die Ableitungsfunktion (Dxf):MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfxMN von ff nach xx als

    (fundefeq[fxa]Dxf(a))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfxahttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa

    Die Abhängigkeit von xx ist bleibt dieser Notation (Lagrange Notation) implizit. In Leibniz Notation schreiben wir die Ableitungsfunktion von ff nach xx als Dxfhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfx. In Eulers Notation schreiben wir Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa und in Newtonnotation Dxf(a)http://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtatfxa.


  • Newton-Mercator-Reihe

    Die Mercator-Reihe oder Newton-Mercator-Reihe ist die Taylorreihe des natürlichen Logarithmus.

    ln((1+x))http://mathhub.info/smglom/calculus/naturallogarithm.omdoc?naturallogarithm?natural-logarithmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionx

    Die Reihe konvergiert für (multi-relation-expression1lethanxlessthan1)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanxhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lessthan.


  • Niven-Zahl en

    Eine Harshad-Zahl oder Niven-Zahl in einer gegebenen Zahlenbasis ist eine ganze Zahl die durch die Summe ihrer Ziffern in dieser Zahlenbasis teilbar ist.


  • Nivens Konstante en

    Man definiert (H1)=1http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationH und Hnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationHn sei der größte Exponent, der bei der Primfaktorzerlegung einer natürlichen Zahl n>1http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethann auftritt.

    Nivens Konstante ist dann gegeben durch

    limn(1n)http://mathhub.info/smglom/calculus/sequenceConvergence.omdoc?sequenceConvergence?limitnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionn

  • normale Zahl zur Basis en

    Wir nennen xx ist eine normale Zahl zur Basis bb, wenn ihre Ziffern, Ziffernpaare, Zifferntripel usw. in der Darstellung zur Basis bb gleich häufig auftreten.


  • NSW-Primzahl , en

    Eine Newman-Shanks-Williams-Primzahl, oder NSW-Primzahl , ist eine Primzahl pp, die in folgender Form geschrieben werden kann

    p=((((1+(2))((2m)+1))+((1-(2))((2m)+1)))2)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?square-roothttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?square-roothttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationm

    Dabei ist mm eine natürliche Zahl.


  • Null en

    Eine Struktur Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR heißt Ring, wenn Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR eine Abelsch e Gruppe ist (die additive Struktur), Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR ein Monoid (die multiplikative Struktur) ist, sowie Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR is a Ringoid.

    Wir nennen 0 die Null des Rings und entsprechend 1 die Eins.


  • nullfrei en

    Eine ganze Zahl Qist nullfrei, wenn ihre Dezimaldarstellung keine Ziffer 0 enthält.


  • Nullstelle en

    Sei KK ein Ring, 0Khttp://mathhub.info/smglom/sets/set.omdoc?set?insetK seine additive Einheit und f:DKhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfDK für eine Menge DD, dann nennen wir jedes z(f-1(0))http://mathhub.info/smglom/sets/set.omdoc?set?insetzhttp://mathhub.info/smglom/sets/image.omdoc?image?pre-imagef eine Nullstelle von ff.


  • obere en

    Wir definieren das Produkt über eine Folge aiOMFOREIGNscala.xml.Node$@6040b6ef durch

    (i=nmai):=(defined-piecewise(
    0wenn(n=m)
    )
    (
    (an)sonst
    )
    )
    http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdfromtonmiOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnmhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionOMFOREIGNscala.xml.Node$@6040b6ef

    Die Variable ii wird der Laufindex oder Multiplikationsindex genannt, nn die untere (oder Startwert) und mm die obereGrenze (oder Endwert) des Produkts, zusammen bestimmen sie den Multiplikationsbereich.

    Gebräuchlich sind auch die fogenden Varianten des Produktoperators: φaihttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdCollφOMFOREIGNscala.xml.Node$@6040b6ef und iSaihttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdInCollSiOMFOREIGNscala.xml.Node$@6040b6ef. Der erste spezifiziert den Multiplikationsbereich durch eine Formel φφ über ii und der zweite gibt den Multiplikationsbereich direkt als eine Menge SS an.


  • obere en

    Summation ist iterierte Addition. Wir definieren die Summe über eine Folge aiOMFOREIGNscala.xml.Node$@6040b6ef durch

    (i=nmai):=(defined-piecewise(
    0wenn(n=m)
    )
    (
    (an)sonst
    )
    )
    http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/numberfields/sum.omdoc?sum?sumnmiOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnmhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionOMFOREIGNscala.xml.Node$@6040b6ef

    Die Variable ii wird der Laufindex oder Summationsindex genannt, nn die untere (oder Startwert) und mm die obereGrenze (oder Endwert) der Summe, zusammen bestimmen sie den Summationsbereich.

    Gebräuchlich sind auch die fogenden Varianten des Summenoperators: (SumCollφ[i]ai)http://mathhub.info/smglom/numberfields/sum.omdoc?sum?SumCollφiOMFOREIGNscala.xml.Node$@6040b6ef und iSaihttp://mathhub.info/smglom/numberfields/sum.omdoc?sum?SumInCollSiOMFOREIGNscala.xml.Node$@6040b6ef. Der erste spezifiziert den Summationsbereich durch eine Formel φφ über ii und der zweite gibt den Summationsbereich als eine Menge SS an.


  • Obermenge Show Notations ro tr en

    Eine Menge AA ist eine Obermenge einer Menge BB (schreibe ABhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?proper-supersetAB), wenn BAhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?proper-subsetBA.


  • offene Kugel Show Notations en

    In einem metrischen Raum M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd nennen wir die Menge (fundefeq[rx]𝐵(r,x))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqrxhttp://mathhub.info/smglom/calculus/ball.omdoc?ball?open-ballrx die offene Kugel nd (fundefeq[rx]B¯(r,x))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqrxhttp://mathhub.info/smglom/calculus/ball.omdoc?ball?closed-ballrx die geschlossene Kugel um xx mit Radius rr. Wir schreiben auch 𝐵(r,x)http://mathhub.info/smglom/calculus/ball.omdoc?ball?open-ballrx und B¯(r,x)http://mathhub.info/smglom/calculus/ball.omdoc?ball?closed-ballrx.


  • offene Menge en

    Ein topologischer Raum X,Ohttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureXOMFOREIGNscala.xml.Node$@6040b6ef ist eine Menge XX zusammen mit einer Familie O(𝒫(X))http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/sets/powerset.omdoc?powerset?powersetX, so daß

    1. 1.

      ,XOhttp://mathhub.info/smglom/sets/set.omdoc?set?minsethttp://mathhub.info/smglom/sets/emptyset.omdoc?emptyset?esetXOMFOREIGNscala.xml.Node$@6040b6ef.

    2. 2.

      (SSs)Ohttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/union.omdoc?union?munionCollectionOMFOREIGNscala.xml.Node$@6040b6efSsOMFOREIGNscala.xml.Node$@6040b6ef falls SShttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetSOMFOREIGNscala.xml.Node$@6040b6ef.

    3. 3.

      (SSs)Ohttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/intersection.omdoc?intersection?mintersectCollectionOMFOREIGNscala.xml.Node$@6040b6efSsOMFOREIGNscala.xml.Node$@6040b6ef falls SShttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetSOMFOREIGNscala.xml.Node$@6040b6ef endlich.

    Dann heißt OOMFOREIGNscala.xml.Node$@6040b6ef eine Topologie) auf XX. Elemente der Topologie OOMFOREIGNscala.xml.Node$@6040b6ef heißen offene Mengen und ihre Komplemente abgeschlossen oder einfach geschlossen . Eine Teilmenge von XX kann weder gesclossen noch offen, oder gesclossen, oder offen oder beides.


  • Ordnung en

    Die Eckenanzahl (Anzahl der Ecken eines Graphen) GG wird seine Ordnung genannt, man schreibt |G|http://mathhub.info/smglom/graphs/graphnumberofvertices.omdoc?graphnumberofvertices?graphnumberofverticesG.


  • Ore-Zahl en

    Eine harmonische-Teiler-Zahl, oder Ore-Zahl, ist eine positive ganze Zahl, deren Harmonisches Mittel ihrer Teiler eine ganze Zahl ist.


  • Paar Show Notations tr en ro

    Die Menge A,Bhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairAB der Paare über den Mengen AA und BB ist definiert als (bsetst[ab]a,b)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstabhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairab. Wir nennen (a,b)(A×B)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairabhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAB ein Paar.


  • Paare Show Notations tr en ro

    Die Menge A,Bhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairAB der Paare über den Mengen AA und BB ist definiert als (bsetst[ab]a,b)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstabhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairab. Wir nennen (a,b)(A×B)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairabhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAB ein Paar.


  • paarweise disjunkt ro en tr

    Eine Familie von Mengen heißt paarweise disjunkt, wenn je zwei Mengen disjunkt sind.


  • Padovan-Zahlen en

    Die Padovan-Zahlen werden rekursiv definiert

    (multi-relation-expressionP0equalP1equalP2equal1)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numbers/padovannumbers.omdoc?padovannumbers?padovannumbershttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/padovannumbers.omdoc?padovannumbers?padovannumbershttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/padovannumbers.omdoc?padovannumbers?padovannumbershttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal
    (Pn)=((P(n-2))+(P(n-3)))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/padovannumbers.omdoc?padovannumbers?padovannumbersnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numbers/padovannumbers.omdoc?padovannumbers?padovannumbershttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numbers/padovannumbers.omdoc?padovannumbers?padovannumbershttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn

  • Partialsumme Show Notations en

    Ist (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan eine Folge mit (an)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselanhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number oder (an)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselanhttp://mathhub.info/smglom/numberfields/complexnumbers.omdoc?complexnumbers?complex-number, so definieren wir die nn-te Partialsumme durch (fundefeq[an]sn)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqanhttp://mathhub.info/smglom/calculus/series.omdoc?series?partial-suman.

    Die durch (na)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequencean induzierte Reihe ist die Partialsummenfolge (sequenceon[an]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonann.


  • Partialsummenfolge en

    Ist (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan eine Folge mit (an)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselanhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number oder (an)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselanhttp://mathhub.info/smglom/numberfields/complexnumbers.omdoc?complexnumbers?complex-number, so definieren wir die nn-te Partialsumme durch (fundefeq[an]sn)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqanhttp://mathhub.info/smglom/calculus/series.omdoc?series?partial-suman.

    Die durch (na)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequencean induzierte Reihe ist die Partialsummenfolge (sequenceon[an]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonann.


  • partielle Funktion en

    Eine Relation f(X×Y)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetfhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsXY, heißt partielle Funktion mit Argumentbereich XX (schreibe 𝐝𝐨𝐦(f)http://mathhub.info/smglom/sets/functions.omdoc?functions?domainf) und Wertebereich YY (schreibe 𝐜𝐨𝐝𝐨𝐦(f)http://mathhub.info/smglom/sets/functions.omdoc?functions?codomainf), wenn es für jedes xXhttp://mathhub.info/smglom/sets/set.omdoc?set?insetxX höchstens ein yYhttp://mathhub.info/smglom/sets/set.omdoc?set?insetyY gibt mit (x,y)fhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairxyf.

    Wir schreiben f:XY;xyhttp://mathhub.info/smglom/sets/functions.omdoc?functions?partfunsuchthatfXYxy und (f)=yhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalfy wenn (x,y)fhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairxyf.


  • partielle Ordnung en

    Eine quasiordnung pole(A×A)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsethttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?polehttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAA auf AA heißt partielle Ordnung, wenn sie antisymmetrisch ist. polehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole induziert eine strikte Ordnung poless:=(bsetst[ab](a,b)pole)http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?polesshttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstabhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairabhttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole.

    Die konversen Relationen schreiben wir oft als pomehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pome und pomorehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pomore.

    Wir nennen eine Struktur S,polehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureShttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole aus einer Menge SS und einer partiellen Ordnung polehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole eine geordnete Menge.

    Eine partielle Ordnung RR heißt totale Ordnung (or einfache Ordnung oder lineare Ordnung), falls abhttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?poleab oder bahttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?poleba für alle a,bAhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabA.

    Wir nennen eine Struktur S,polehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureShttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole aus einer Menge SS und einer totalen Ordnung polehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole eine total geordnete Menge.


  • Pell-Zahlen en

    Die Pell-Zahlen werden rekursiv definiert mit den Gleichungen

    P0http://mathhub.info/smglom/numbers/pellnumbers.omdoc?pellnumbers?pellnumbers equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal 0
    P1http://mathhub.info/smglom/numbers/pellnumbers.omdoc?pellnumbers?pellnumbers equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal 1
    Pnhttp://mathhub.info/smglom/numbers/pellnumbers.omdoc?pellnumbers?pellnumbersn equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal (2(P(n-1)))+(P(n-2))http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numbers/pellnumbers.omdoc?pellnumbers?pellnumbershttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numbers/pellnumbers.omdoc?pellnumbers?pellnumbershttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn

  • perfekte Zahl en

    Eine natürliche Zahl nn wird vollkommene Zahl (auch perfekte Zahl) genannt, wenn sie gleich der Summe aller ihrer (positiven) echten Teiler ist.


  • permutierbare Primzahl en

    Eine permutierbare Primzahl ist eine Primzahl, die bei jeder Permutation ihrer Ziffern wieder eine Primzahl ist.

    Zum Beispiel:

    131

  • Perrin-Pseudoprimzahl en

    Eine Perrin-Pseudoprimzahl ist eine zusammengesetzte Zahl nn, die die Perrin-Zahl Pnhttp://mathhub.info/smglom/numbers/perrinpseudoprime.omdoc?perrinpseudoprime?perrinpseudoprimen teilt.


  • Perrin-Zahlen en

    Die Perrin-Zahlen werden rekursiv definiert:

    P0http://mathhub.info/smglom/numbers/perrinnumbers.omdoc?perrinnumbers?perrinnumbers equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal 3
    P1http://mathhub.info/smglom/numbers/perrinnumbers.omdoc?perrinnumbers?perrinnumbers equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal 0
    P2http://mathhub.info/smglom/numbers/perrinnumbers.omdoc?perrinnumbers?perrinnumbers equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal 2
    Pnhttp://mathhub.info/smglom/numbers/perrinnumbers.omdoc?perrinnumbers?perrinnumbersn equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal (P(n-2))+(P(n-3))http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numbers/perrinnumbers.omdoc?perrinnumbers?perrinnumbershttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numbers/perrinnumbers.omdoc?perrinnumbers?perrinnumbershttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn

  • Pfad Show Notations en

    Ist G=(V,E)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalGhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE gerichteter Graph, so nennen wir einen Vektor (p=((v0,,vn)))(V(n+1))http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalphttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlivnhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?nCartSpaceVhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionn einen Pfad in GG wenn (vi-1,vi)Ehttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6efE für alle 1inhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?betweeneein mit n>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethann.

    • v0OMFOREIGNscala.xml.Node$@6040b6ef heißt der Startknoten von pp (schreibe start(p)http://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pstartp )

    • vnOMFOREIGNscala.xml.Node$@6040b6ef heißt der Endknoten von pp (schreibe end(p)http://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pendp )

    • nn heißt die Länge von pp (schreibe len(p)http://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?plenp )


  • Pfeile en

    Ein Graph ist ein Paar V,Ehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE, so dass VV eine Menge ist und E(V,V)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetEhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairVV eine Teilmenge der Menge der Paare aus VV. Wir nennen VV die Ecken (oder Punkte, Knoten) and EE die Kanten (oder Linien, Pfeile) von GG.


  • Pierpont-Primzahl en

    Eine Pierpont-Primzahl ist eine Primzahl der Form ((2u)(3v))+1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationuhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationv mit nichtnegativen ganzen Zahlen uu und vv.


  • Plastic-Konstante Show Notations en

    Die Plastic-Zahl ρhttp://mathhub.info/smglom/smglom/plasticnumber.omdoc?plasticnumber?plastic-number (auch Plastic-Konstante genannt) ist die einzige reelle Lösung der kubischen Gleichung (x3)=(x+1)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionx Ihr Wert beträgt:

    ρ=((((108+(12(69)))3)+((108-(12(69)))3))6)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/smglom/plasticnumber.omdoc?plasticnumber?plastic-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?roothttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?square-roothttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?roothttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?square-root
    ρ=1.324717957244746...http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?startswithρ

  • Plastic-Zahl Show Notations en

    Die Plastic-Zahl ρhttp://mathhub.info/smglom/smglom/plasticnumber.omdoc?plasticnumber?plastic-number (auch Plastic-Konstante genannt) ist die einzige reelle Lösung der kubischen Gleichung (x3)=(x+1)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationxhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionx Ihr Wert beträgt:

    ρ=((((108+(12(69)))3)+((108-(12(69)))3))6)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/smglom/plasticnumber.omdoc?plasticnumber?plastic-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?roothttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?square-roothttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?roothttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?square-root
    ρ=1.324717957244746...http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?startswithρ

  • positiven natürlichen Zahlen Show Notations en

    Die Menge http://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?positive-natural-number der positiven natürlichen Zahlen ist {(1,2,3,)}http://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?aseqdots.


  • positiven rationalen Zahlen Show Notations

    Die Menge http://mathhub.info/smglom/numberfields/rationalnumbers.omdoc?rationalnumbers?rational-numbers der rationalen Zahlen ist die Menge (bsetst[nm]nm)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstnmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionnm.

    Wir schreiben http://mathhub.info/smglom/numberfields/rationalnumbers.omdoc?rationalnumbers?negative-rational-numbers und http://mathhub.info/smglom/numberfields/rationalnumbers.omdoc?rationalnumbers?positive-rational-numbers für die Mengen der negativen rationalen Zahlen und der positiven rationalen Zahlen.


  • positiven rationalen Zahlen Show Notations

    Die Menge http://mathhub.info/smglom/numberfields/rationalnumbers.omdoc?rationalnumbers?rational-numbers der rationalen Zahlen ist die Menge (bsetst[nm]nm)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstnmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionnm.

    Wir schreiben http://mathhub.info/smglom/numberfields/rationalnumbers.omdoc?rationalnumbers?negative-rational-numbers und http://mathhub.info/smglom/numberfields/rationalnumbers.omdoc?rationalnumbers?positive-rational-numbers für die Mengen der negativen rationalen Zahlen und der positiven rationalen Zahlen.


  • positiven reellen Zahlen Show Notations en

    Die Menge http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number der reellen Zahlen reelleZahlen ist die Vervollständigung von http://mathhub.info/smglom/numberfields/rationalnumbers.omdoc?rationalnumbers?rational-numbers.

    Wir schreiben http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?negative-real-number und http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?positive-real-number für die Mengen der negativen reellen Zahlen und der positiven reellen Zahlen.


  • positiven reellen Zahlen Show Notations en

    Die Menge http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number der reellen Zahlen reelleZahlen ist die Vervollständigung von http://mathhub.info/smglom/numberfields/rationalnumbers.omdoc?rationalnumbers?rational-numbers.

    Wir schreiben http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?negative-real-number und http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?positive-real-number für die Mengen der negativen reellen Zahlen und der positiven reellen Zahlen.


  • potente Zahl en

    Eine potente Zahl ist eine natürliche Zahl mm, so dass für jeden Primteiler pp von mm auch p2http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationp Teiler von mm ist.

    Eine potente Zahl ist das Produkt einer Quadratzahl und einer Kubikzahl.


  • Potenzmenge Show Notations ro tr bg

    Für eine Menge AA definieren wir die Potenzmenge 𝒫(A)http://mathhub.info/smglom/sets/powerset.omdoc?powerset?powersetA von AA als (bsetst[S]S)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstSS.


  • Poulet-Zahlen en

    Poulet-Zahlen sind Fermatsche fermatpseudoprime zur Basis 2.


  • praktische Zahl en

    Eine praktische Zahl ist eine positive ganze Zahl nn, so dass alle kleineren positiven ganzen Zahlen als Summe verschiedener Teiler von nn dargestellt werden können.


  • Primfaktor en

    Eine Primzahl pp ist ein Primfaktor der natürlichen Zahl nn, wenn pp ein Teiler von nn ist.


  • primitiv en

    Ein pythagoreisches Zahlentripel besteht aus drei positiven ganzen Zahlen aa, bb, und cc, für die gilt ((a2)+(b2))=(c2)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationahttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationbhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationc.

    Ein pythagoreisches Tripel heißt primitiv wenn aa, bb und cc teilerfremd sind.


  • Primitivwurzel en

    Eine Zahl gg heißt Primitivwurzel modulo nn, wenn jede teilerfremde Zahl nn kongruent modulo nn zu einer Potenz von gg ist.


  • Primorial Show Notations en

    Das Primorial pnhttp://mathhub.info/smglom/smglom/primorial.omdoc?primorial?primorialn einer natürlichen Zahl nn ist das Produkt aller Primzahlen kleiner oder gleich dieser Zahl.


  • Primzahl Show Notations en

    Eine Primzahl ist eine natürliche Zahl, die genau zwei natürliche Zahlen als Teiler hat. Die Anzahl der Primzahlen kleiner oder gleich nn bezeichnet man mit π(n)http://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?NumberPrimeNumbern.


  • Primzahlfünfling en

    Wenn {p,(p+2),(p+6),(p+8)}http://mathhub.info/smglom/sets/set.omdoc?set?setphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionp ein Primzahlvierling ist und p-4http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionp oder p+12http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionp auch eine Primzahl ist, dann sind die fünf Primzahlen ein Primzahlfünfling.

    Zum Beispiel:

    {5,7,11,13,17}http://mathhub.info/smglom/sets/set.omdoc?set?set
    {7,11,13,17,19}http://mathhub.info/smglom/sets/set.omdoc?set?set
    {11,13,17,19,23}http://mathhub.info/smglom/sets/set.omdoc?set?set

  • Primzahllücke Show Notations en

    Eine Primzahllücke ist die Differenz zwischen zwei aufeinanderfolgenden Primzahlen.

    (gn)=((p(n+1))-(pn))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/primes/primegap.omdoc?primegap?prime-gapnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?prime-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnhttp://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?prime-numbern

  • Primzahlsechsling en

    Wenn {p,(p+2),(p+6),(p+8)}http://mathhub.info/smglom/sets/set.omdoc?set?setphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionp ein Primzahlvierling ist und p-4http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionp und p+12http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionp auch Primzahlen sind, dann sind die sechs Primzahlen ein Primzahlsechsling.

    Zum Beispiel:

    {7,11,13,17,19,23}http://mathhub.info/smglom/sets/set.omdoc?set?set
    {97,101,103,107,109,113}http://mathhub.info/smglom/sets/set.omdoc?set?set
    {16057,16061,16063,16067,16069,16073}http://mathhub.info/smglom/sets/set.omdoc?set?set

  • Primzahlvierling en

  • Primzahlzwilling en

    Ein Primzahlzwilling ist ein Paar von Primzahlen mit Differenz 2.


  • Produkt en

    Wir definieren das Produkt über eine Folge aiOMFOREIGNscala.xml.Node$@6040b6ef durch

    (i=nmai):=(defined-piecewise(
    0wenn(n=m)
    )
    (
    (an)sonst
    )
    )
    http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdfromtonmiOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnmhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionOMFOREIGNscala.xml.Node$@6040b6ef

    Die Variable ii wird der Laufindex oder Multiplikationsindex genannt, nn die untere (oder Startwert) und mm die obereGrenze (oder Endwert) des Produkts, zusammen bestimmen sie den Multiplikationsbereich.

    Gebräuchlich sind auch die fogenden Varianten des Produktoperators: φaihttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdCollφOMFOREIGNscala.xml.Node$@6040b6ef und iSaihttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdInCollSiOMFOREIGNscala.xml.Node$@6040b6ef. Der erste spezifiziert den Multiplikationsbereich durch eine Formel φφ über ii und der zweite gibt den Multiplikationsbereich direkt als eine Menge SS an.


  • Projektion Show Notations en

    Sei AA eine Familie von Mengen dann definieren wir das nn-fache Cartesische Product A1×...×Anhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?ncartliAn als (bsetst[an](a1,,an))http://mathhub.info/smglom/sets/set.omdoc?set?bsetstanhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlian, und nennen ((a1,,an))(A1×...×An)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlianhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?ncartliAn nn-Tupel.

    Wir nennen die Funktion (projectioni):(A1×...×An)Ai;((a1,,an))aihttp://mathhub.info/smglom/sets/functions.omdoc?functions?funsuchthathttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?projectionihttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?ncartliAnOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlianOMFOREIGNscala.xml.Node$@6040b6ef die iite Projektion.


  • Projektion Show Notations en

    Seien SS eine Menge, RR eine Äquivalenzrelation auf SS und xShttp://mathhub.info/smglom/sets/set.omdoc?set?insetxS, dann nennen wir die Menge (fundefeq[xR][x]R)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqxRhttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?equivalence-classxR die Äquivalenzklasse von xx (unter RR), und die Menge (fundefeq[SR]S_R)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqSRhttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?quotient-spaceSR die Quotientenmenge von SS (unter RR).

    Wir nennen die Abbildung (projectionR):S(S_R);x([x]R)http://mathhub.info/smglom/sets/functions.omdoc?functions?funsuchthathttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?projectionRShttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?quotient-spaceSRxhttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?equivalence-classxR die Projektion von SS auf S_Rhttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?quotient-spaceSR unter RR.


  • pronische Zahl Show Notations en

    Eine pronische Zahl Pnhttp://mathhub.info/smglom/numbers/pronicnumber.omdoc?pronicnumber?pronic-numbern, Rechteckzahl, oder Rechteckszahl ist eine natürliche Zahl, die das Produkt zweier aufeinanderfolgender natürlicher Zahlen ist.

    (multi-relation-expressionPnequal(nn+1)equal(n2)+n)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numbers/pronicnumber.omdoc?pronicnumber?pronic-numbernhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationnn

  • Proth-Zahl en

    Eine Proth-Zahl ist eine natürlicheZahl der Form (k(2n))+1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationkhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn wobei kk eine ungerade und nn eine positive ganze Zahl mit (2n)>khttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationnk ist.

    Die Cullen-Zahlen (n2n)+1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn und die Fermat-Zahlen (2(2n))+1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn sind Spezialfälle der Proth-Zahlen.


  • Proth-Zahlen en

    Eine Proth-Zahl ist eine natürlicheZahl der Form (k(2n))+1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationkhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn wobei kk eine ungerade und nn eine positive ganze Zahl mit (2n)>khttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationnk ist.

    Die Cullen-Zahlen (n2n)+1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn und die Fermat-Zahlen (2(2n))+1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn sind Spezialfälle der Proth-Zahlen.


  • Prothsche Primzahl en

    Eine Prothsche Primzahl ist eine Primzahl der Form (k2n)+1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionkhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn wobei kk eine ungerade und nn eine positive ganze Zahl mit (2n)>khttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationnk ist.


  • Punkte en

    Ein Graph ist ein Paar V,Ehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE, so dass VV eine Menge ist und E(V,V)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetEhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairVV eine Teilmenge der Menge der Paare aus VV. Wir nennen VV die Ecken (oder Punkte, Knoten) and EE die Kanten (oder Linien, Pfeile) von GG.


  • Pythagoras-Box en

    Eine Pythagoras-Box ist ein Quadrupel a,b,c,dhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tupleabcd von natürlichen Zahlen, die einen Quader mit den Seitenlängen aa, bb, cc und einer Raumdiagonalen der Länge dd definieren.


  • pythagoreisches Zahlentripel en

    Ein pythagoreisches Zahlentripel besteht aus drei positiven ganzen Zahlen aa, bb, und cc, für die gilt ((a2)+(b2))=(c2)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationahttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationbhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationc.

    Ein pythagoreisches Tripel heißt primitiv wenn aa, bb und cc teilerfremd sind.


  • pythagoräische Primzahl en

    Eine pythagoräische Primzahl ist eine Primzahl der Form (4n)+1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationn.


  • Pythagoräisches Quadrupel en

    Ein Pythagoräisches Quadrupel (a,b,c,d)http://mathhub.info/smglom/smglom/pythagoreanquadruple.omdoc?pythagoreanquadruple?pythquadabcd ist ein Tupel von vier ganzen Zahlen aa, bb, cc und d>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethand, für das gilt

    ((a2)+(b2)+(c2))=(d2)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationahttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationbhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationchttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationd

  • quadratfrei

    Eine natürliche Zahl heißt quadratfrei, wenn in ihrer Primfaktorzerlegung keine Primzahl mehr als einmal auftritt.


  • quadratische Mittel Show Notations en

    Das quadratische Mittel einer Menge {x1,x2,,xn}http://mathhub.info/smglom/sets/set.omdoc?set?setOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6ef ist definiert als:

    =(((1n)((x12)+(x22)++(xn2))))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/smglom/rootmeansquare.omdoc?rootmeansquare?root-mean-squarehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?square-roothttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationOMFOREIGNscala.xml.Node$@6040b6ef

  • Quelle

    Sei G=(V,E)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalGhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE ein gerichteter Graph, dann nennen wir einen Knoten vVhttp://mathhub.info/smglom/sets/set.omdoc?set?insetvV

    • initial (oder eine Quelle) in GG, wenn es kein wVhttp://mathhub.info/smglom/sets/set.omdoc?set?insetwV gibt, so dass (w,v)Ehttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairwvE.

    • terminal (oder Senke) in GG, wenn es kein wVhttp://mathhub.info/smglom/sets/set.omdoc?set?insetwV gibt, so dass (v,w)Ehttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairvwE.


  • Querprodukt en

    Das Querprodukt einer natürlichen Zahl ist das Produkt ihrer Ziffern.


  • Quersumme en

    Als Quersumme bezeichnet man die Summe der Ziffern einer natürlichen Zahl.


  • Quotientenmenge Show Notations en

    Seien SS eine Menge, RR eine Äquivalenzrelation auf SS und xShttp://mathhub.info/smglom/sets/set.omdoc?set?insetxS, dann nennen wir die Menge (fundefeq[xR][x]R)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqxRhttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?equivalence-classxR die Äquivalenzklasse von xx (unter RR), und die Menge (fundefeq[SR]S_R)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqSRhttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?quotient-spaceSR die Quotientenmenge von SS (unter RR).

    Wir nennen die Abbildung (projectionR):S(S_R);x([x]R)http://mathhub.info/smglom/sets/functions.omdoc?functions?funsuchthathttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?projectionRShttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?quotient-spaceSRxhttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?equivalence-classxR die Projektion von SS auf S_Rhttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?quotient-spaceSR unter RR.


  • Ramanujan-6-10-8-Identität en

    Sei (ad)=(bc)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationadhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationbc für reelle Zahlen aa, bb, cc und dd, dann ist die Ramanujan-6-10-8-Identität gegeben durch

    ((64((((((a+b+c)6)+((b+c+d)6))-((c+d+a)6)-((d+a+b)6))+(((a)d)6))-(((b)c)6)))((((((a+b+c)10)+((b+c+d)10))-((c+d+a)10)-((d+a+b)10))+(((a)d)10))-(((b)c)10)))=(45(((((((a+b+c)8)+((b-c-d)8))-((c-d-a)8)-((d+a+b)8))+(((a)d)8))-(((b)c)8))2))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionabchttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionbcdhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additioncdahttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additiondabhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionadhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionbchttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionabchttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionbcdhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additioncdahttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additiondabhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionadhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionbchttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionabchttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionbcdhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractioncdahttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additiondabhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionadhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionbc

  • Ramanujan-Konstante Show Notations en

  • Ramanujan-Soldner-Konstante Show Notations en

  • rationalen Zahlen Show Notations

    Die Menge http://mathhub.info/smglom/numberfields/rationalnumbers.omdoc?rationalnumbers?rational-numbers der rationalen Zahlen ist die Menge (bsetst[nm]nm)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstnmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionnm.

    Wir schreiben http://mathhub.info/smglom/numberfields/rationalnumbers.omdoc?rationalnumbers?negative-rational-numbers und http://mathhub.info/smglom/numberfields/rationalnumbers.omdoc?rationalnumbers?positive-rational-numbers für die Mengen der negativen rationalen Zahlen und der positiven rationalen Zahlen.


  • rationalen Zahlen Show Notations

    Die Menge http://mathhub.info/smglom/numberfields/rationalnumbers.omdoc?rationalnumbers?rational-numbers der rationalen Zahlen ist die Menge (bsetst[nm]nm)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstnmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionnm.

    Wir schreiben http://mathhub.info/smglom/numberfields/rationalnumbers.omdoc?rationalnumbers?negative-rational-numbers und http://mathhub.info/smglom/numberfields/rationalnumbers.omdoc?rationalnumbers?positive-rational-numbers für die Mengen der negativen rationalen Zahlen und der positiven rationalen Zahlen.


  • Raum der Funktionen Show Notations en

    Wir nennen die Menge ABhttp://mathhub.info/smglom/sets/functions.omdoc?functions?function-spaceAB aller Funktionen von AA nach BB den Raum der Funktionen von AA nach BB.

    Analog ist ABhttp://mathhub.info/smglom/sets/functions.omdoc?functions?partial-function-spaceAB der Raum der partiellen Funktionen von AA nach BB.


  • Raum der partiellen Funktionen Show Notations

    Wir nennen die Menge ABhttp://mathhub.info/smglom/sets/functions.omdoc?functions?function-spaceAB aller Funktionen von AA nach BB den Raum der Funktionen von AA nach BB.

    Analog ist ABhttp://mathhub.info/smglom/sets/functions.omdoc?functions?partial-function-spaceAB der Raum der partiellen Funktionen von AA nach BB.


  • Raum der partiellen Funktionen Show Notations

    Wir nennen die Menge ABhttp://mathhub.info/smglom/sets/functions.omdoc?functions?function-spaceAB aller Funktionen von AA nach BB den Raum der Funktionen von AA nach BB.

    Analog ist ABhttp://mathhub.info/smglom/sets/functions.omdoc?functions?partial-function-spaceAB der Raum der partiellen Funktionen von AA nach BB.


  • Rechteckszahl Show Notations en

    Eine pronische Zahl Pnhttp://mathhub.info/smglom/numbers/pronicnumber.omdoc?pronicnumber?pronic-numbern, Rechteckzahl, oder Rechteckszahl ist eine natürliche Zahl, die das Produkt zweier aufeinanderfolgender natürlicher Zahlen ist.

    (multi-relation-expressionPnequal(nn+1)equal(n2)+n)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numbers/pronicnumber.omdoc?pronicnumber?pronic-numbernhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationnn

  • Rechteckzahl Show Notations en

    Eine pronische Zahl Pnhttp://mathhub.info/smglom/numbers/pronicnumber.omdoc?pronicnumber?pronic-numbern, Rechteckzahl, oder Rechteckszahl ist eine natürliche Zahl, die das Produkt zweier aufeinanderfolgender natürlicher Zahlen ist.

    (multi-relation-expressionPnequal(nn+1)equal(n2)+n)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numbers/pronicnumber.omdoc?pronicnumber?pronic-numbernhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationnn

  • rechten Teiler en

    Sei Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR ein Ring und a,bRhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabR mit a0http://mathhub.info/smglom/mv/equal.omdoc?equal?notequala, dann nennen wir aa einen linken Teiler von bb (und sagen aa teilt bb links oder bb ist link teilbar durch aa) und bb ein linkes Vielfaches von aa, wenn es ein nRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnR gibt, für die (multiplication)=bhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationb ist.

    Analog nennen wir aa einen rechten Teiler von bb (und sagen aa teilt bb rechts oder bb ist rechts teilbar durch aa) und bb ein rechtes Vielfaches von aa, wenn es ein nRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnR gibt, für die (na)bhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnab ist.

    Ist RR kommutativ, so fallen rechte und linke Teiler zusammen, und wir sprechen einfach von Teiler, teilbar, und Vielfachen. Wir schreiben dann a|bhttp://mathhub.info/smglom/algebra/ring-divisor.omdoc?ring-divisor?divisorab.

    Wir nennen 1, 1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction, nn und nhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn triviale divisoren von nn. Ein Teiler von nn, der nicht trivial ist, heißt echter Teiler von nn.


  • rechtes Vielfaches en

    Sei Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR ein Ring und a,bRhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabR mit a0http://mathhub.info/smglom/mv/equal.omdoc?equal?notequala, dann nennen wir aa einen linken Teiler von bb (und sagen aa teilt bb links oder bb ist link teilbar durch aa) und bb ein linkes Vielfaches von aa, wenn es ein nRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnR gibt, für die (multiplication)=bhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationb ist.

    Analog nennen wir aa einen rechten Teiler von bb (und sagen aa teilt bb rechts oder bb ist rechts teilbar durch aa) und bb ein rechtes Vielfaches von aa, wenn es ein nRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnR gibt, für die (na)bhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnab ist.

    Ist RR kommutativ, so fallen rechte und linke Teiler zusammen, und wir sprechen einfach von Teiler, teilbar, und Vielfachen. Wir schreiben dann a|bhttp://mathhub.info/smglom/algebra/ring-divisor.omdoc?ring-divisor?divisorab.

    Wir nennen 1, 1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction, nn und nhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn triviale divisoren von nn. Ein Teiler von nn, der nicht trivial ist, heißt echter Teiler von nn.


  • rechts teilbar en

    Sei Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR ein Ring und a,bRhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabR mit a0http://mathhub.info/smglom/mv/equal.omdoc?equal?notequala, dann nennen wir aa einen linken Teiler von bb (und sagen aa teilt bb links oder bb ist link teilbar durch aa) und bb ein linkes Vielfaches von aa, wenn es ein nRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnR gibt, für die (multiplication)=bhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationb ist.

    Analog nennen wir aa einen rechten Teiler von bb (und sagen aa teilt bb rechts oder bb ist rechts teilbar durch aa) und bb ein rechtes Vielfaches von aa, wenn es ein nRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnR gibt, für die (na)bhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnab ist.

    Ist RR kommutativ, so fallen rechte und linke Teiler zusammen, und wir sprechen einfach von Teiler, teilbar, und Vielfachen. Wir schreiben dann a|bhttp://mathhub.info/smglom/algebra/ring-divisor.omdoc?ring-divisor?divisorab.

    Wir nennen 1, 1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction, nn und nhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn triviale divisoren von nn. Ein Teiler von nn, der nicht trivial ist, heißt echter Teiler von nn.


  • rechtsseitigen Grenzwert Show Notations en

    Sei f:SThttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfST gegeben mit S,Thttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?msseteqSThttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number und ahttp://mathhub.info/smglom/sets/set.omdoc?set?insetahttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number.

    Dann ist der linksseitige Grenzwert von ff in aa (auch: der Grenzwert von ff wenn xx von unten gegen pp strebt) definiert als diejenige Zahl lhttp://mathhub.info/smglom/sets/set.omdoc?set?insetlhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number, so dass für jedes ϵ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanϵ ein δ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanδ gibt so dass (|(((fx))l)|)ϵhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/numberfields/absolutevalue.omdoc?absolutevalue?absolute-valuehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationfxlϵ für alle (multi-relation-expression0lethan(a)xlethanδ)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionaxhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanδ. Der linksseitige Grenzwert wird geschrieben als limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?leftsided-limitaxf, limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?leftsided-limitaxf oder limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?leftsided-limitaxf.

    Analog definieren wir den rechtsseitigen Grenzwert von ff in aa (auch: der Grenzwert von ff wenn xx von unten gegen pp strebt) definiert als diejenige Zahl lhttp://mathhub.info/smglom/sets/set.omdoc?set?insetlhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number, so dass für jedes ϵ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanϵ ein δ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanδ gibt so dass (|(((f))l)|)ϵhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/numberfields/absolutevalue.omdoc?absolutevalue?absolute-valuehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionflϵ für alle (multi-relation-expression0lethan(x)alethanδ)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionxahttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanδ. Der linksseitige Grenzwert wird geschrieben als limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?rightsided-limitaxf, limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?rightsided-limitaxf oder limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?rightsided-limitaxf.


  • rechtsseitigen Grenzwert Show Notations en

    Sei f:SThttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfST gegeben mit S,Thttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?msseteqSThttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number und ahttp://mathhub.info/smglom/sets/set.omdoc?set?insetahttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number.

    Dann ist der linksseitige Grenzwert von ff in aa (auch: der Grenzwert von ff wenn xx von unten gegen pp strebt) definiert als diejenige Zahl lhttp://mathhub.info/smglom/sets/set.omdoc?set?insetlhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number, so dass für jedes ϵ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanϵ ein δ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanδ gibt so dass (|(((fx))l)|)ϵhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/numberfields/absolutevalue.omdoc?absolutevalue?absolute-valuehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationfxlϵ für alle (multi-relation-expression0lethan(a)xlethanδ)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionaxhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanδ. Der linksseitige Grenzwert wird geschrieben als limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?leftsided-limitaxf, limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?leftsided-limitaxf oder limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?leftsided-limitaxf.

    Analog definieren wir den rechtsseitigen Grenzwert von ff in aa (auch: der Grenzwert von ff wenn xx von unten gegen pp strebt) definiert als diejenige Zahl lhttp://mathhub.info/smglom/sets/set.omdoc?set?insetlhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number, so dass für jedes ϵ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanϵ ein δ>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanδ gibt so dass (|(((f))l)|)ϵhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/numberfields/absolutevalue.omdoc?absolutevalue?absolute-valuehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionflϵ für alle (multi-relation-expression0lethan(x)alethanδ)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionxahttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanδ. Der linksseitige Grenzwert wird geschrieben als limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?rightsided-limitaxf, limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?rightsided-limitaxf oder limxa(f)http://mathhub.info/smglom/calculus/onesidedlimit.omdoc?onesidedlimit?rightsided-limitaxf.


  • Rechtstrunkierbare Primzahlen en

    Linkstrunkierbare Primzahlen sind Primzahlen, in denen an keiner Stelle die Ziffer Null steht und bei denen das Weglassen einer beliebigen Anzahl führender Stellen wieder zu einer Primzahl führt.

    Zum Beispiel: 937

    Beidseitig trunkierbare Primzahlen sind sowohl links- als auch rechtstrunkierbare Primzahlen.

    Zum Beispiel: 3137

    Rechtstrunkierbare Primzahlen sind Primzahlen, bei denen das Weglassen einer beliebigen Anzahl der letzten Stellen wieder zu einer Primzahl führt.

    Zum Beispiel: 3793


  • reellen Zahlen reelle Show Notations en

    Die Menge http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number der reellen Zahlen reelleZahlen ist die Vervollständigung von http://mathhub.info/smglom/numberfields/rationalnumbers.omdoc?rationalnumbers?rational-numbers.

    Wir schreiben http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?negative-real-number und http://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?positive-real-number für die Mengen der negativen reellen Zahlen und der positiven reellen Zahlen.


  • reflexiv en

    Eine Relation R(A×A)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetRhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAA heißt

    • reflexiv auf AA, wenn (a,a)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairaaR für alle aAhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaA, und

    • irreflexiv (or antireflexiv) auf AA, wenn (a,a)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?ninsethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairaaR für alle aAhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaA.


  • regulär en

    Wenn in einem Graph GG alle Ecken denselben Grad aufweisen, sagen wir kk, dann nennt man GG kk-regulär, oder schlicht regulär.

    Ein 3-regulärer Graph wird auch kubischer Graph genannt.


  • Reguläre Zahlen en

    Reguläre Zahlen sind natürliche Zahlen, die Teiler einer Potenz von 60 sind.


  • Reihe en

    Ist (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan eine Folge mit (an)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselanhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number oder (an)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselanhttp://mathhub.info/smglom/numberfields/complexnumbers.omdoc?complexnumbers?complex-number, so definieren wir die nn-te Partialsumme durch (fundefeq[an]sn)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqanhttp://mathhub.info/smglom/calculus/series.omdoc?series?partial-suman.

    Die durch (na)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequencean induzierte Reihe ist die Partialsummenfolge (sequenceon[an]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonann.


  • Relation en

    Eine Menge R(A×B)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetRhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAB heißt (binäre) Relation zwischen AA and BB. Ist A=Bhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalAB so nennen wir RR eine Relation auf AA.


  • Relation auf en

    Eine Menge R(A×B)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetRhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAB heißt (binäre) Relation zwischen AA and BB. Ist A=Bhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalAB so nennen wir RR eine Relation auf AA.


  • relativ prim Show Notations en

    Zwei natürliche Zahlen sind genau dann teilerfremd oder relativ prim, wenn deren größter gemeinsamer Teiler 1 ist.


  • Repunit Zahlen Show Notations

    Eine Repunit-Zahl besteht nur aus Einsen.

    Die Repunit Zahlen zur Basis bb erhält man durch (Rn(b))=(((bn)-1)(b-1))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/repunit.omdoc?repunit?repunitbnbhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationbnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionb für b2http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methanb und n1http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methann.

    Eine Repunit-Primzahl ist eine Repunit-Zahl, die auch Primzahl ist.


  • Repunit-Primzahl en

    Eine Repunit-Zahl besteht nur aus Einsen.

    Die Repunit Zahlen zur Basis bb erhält man durch (Rn(b))=(((bn)-1)(b-1))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/repunit.omdoc?repunit?repunitbnbhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationbnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionb für b2http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methanb und n1http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methann.

    Eine Repunit-Primzahl ist eine Repunit-Zahl, die auch Primzahl ist.


  • Repunit-Zahl Show Notations en

    Eine Repunit-Zahl besteht nur aus Einsen.

    Die Repunit Zahlen zur Basis bb erhält man durch (Rn(b))=(((bn)-1)(b-1))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/repunit.omdoc?repunit?repunitbnbhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationbnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionb für b2http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methanb und n1http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methann.

    Eine Repunit-Primzahl ist eine Repunit-Zahl, die auch Primzahl ist.


  • Riemann’sche Vermutung en

  • Riesel-Zahl en

  • Ring en

    Eine Struktur Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR heißt Ring, wenn Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR eine Abelsch e Gruppe ist (die additive Struktur), Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR ein Monoid (die multiplikative Struktur) ist, sowie Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR is a Ringoid.

    Wir nennen 0 die Null des Rings und entsprechend 1 die Eins.


  • Ring en

    Ein Ring ist ein Halbring Shttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureS, so dass Shttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureS eine Abelsche Gruppe ist.


  • Sarrus-Zahlen en

    Sarrus-Zahlen sind Fermatsche pseudo-Primzahl zur Basis 2.


  • Schleife en

    Ist G=(V,E)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalGhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE ein gerichteter Graph, so nennen wir

    • einen Pfad pp in GG zyklisch (auch einen Zykel oder eine Schleife), falls (start(p))=(end(p))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pstartphttp://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pendp.

    • einen Zykel (v0,,vn)http://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlivn einfach, wenn vivjhttp://mathhub.info/smglom/mv/equal.omdoc?equal?notequalOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6ef für alle i,j1nhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?mbetweeneeijn mit ijhttp://mathhub.info/smglom/mv/equal.omdoc?equal?notequalij.

    • GG azyklisch, wenn GG keinen Zykel enthält.


  • Schnapszahl en

    Eine Schnapszahl (repdigit) ist eine mehrstellige Zahl, die ausschließlich durch identische Ziffern dargestellt wird.

    Die Darstellung der Schnapszahlen zur Basis bb ist (m((bn)-1)(b-1))mhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationbnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionb für b2http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methanb, n1http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?methann und (multi-relation-expression0lethanmlethanb)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanmhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanb.


  • Schnitt Show Notations tr ro en

    Ist SS eine Familie von Mengen, so ist der Schnitt iISihttp://mathhub.info/smglom/sets/intersection.omdoc?intersection?mintersectCollectionIiOMFOREIGNscala.xml.Node$@6040b6ef über SS gegeben als (bsetst[x]x)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstxx.


  • Schröder-Hipparchus-Zahl Show Notations en

    Die Schröder-Hipparchus-Zahl (oder super-Catalan-Zahl) S(n)http://mathhub.info/smglom/numbers/schroederhipparchusnumberformula.omdoc?schroederhipparchusnumberformula?Schroeder-Hipparchus-numbern ist für natürliche Zahl nn definiert als

    S(n)http://mathhub.info/smglom/numbers/schroederhipparchusnumberformula.omdoc?schroederhipparchusnumberformula?Schroeder-Hipparchus-numbern

  • Schröder-Hipparchus-Zahlen en

    Die Schröder-Hipparchus-Zahlen (oder super-Catalan-Zahlen ) S(n)http://mathhub.info/smglom/numbers/schroederhipparchusnumberrec.omdoc?schroederhipparchusnumberrec?schroederhipprecn werden für natürliche Zahl nn rekursiv definiert:

    S(1)http://mathhub.info/smglom/numbers/schroederhipparchusnumberrec.omdoc?schroederhipparchusnumberrec?schroederhipprec equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal (S(2))=1http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/schroederhipparchusnumberrec.omdoc?schroederhipparchusnumberrec?schroederhipprec
    S(n)http://mathhub.info/smglom/numbers/schroederhipparchusnumberrec.omdoc?schroederhipparchusnumberrec?schroederhipprecn equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal (1n)((((6n)-9)(S((n-1))))-((n-3)(S((n-2)))))http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnhttp://mathhub.info/smglom/numbers/schroederhipparchusnumberrec.omdoc?schroederhipparchusnumberrec?schroederhipprechttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numbers/schroederhipparchusnumberrec.omdoc?schroederhipparchusnumberrec?schroederhipprechttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn

  • Schröder-Zahl en

    Eine Schröder-Zahl ist die Anzahl der Wege von der südwest-Ecke 0,0http://mathhub.info/smglom/sets/pair.omdoc?pair?pair eines quadratischen nnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnn Gitters zur nordost-Ecke n,nhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairnn, unter Verwendung von Einzelschritten in Richtung Nord, Nordost oder Ost ohne die SW-NO Diagonale zu überschreiten.

    Die ersten Schröder-Zahlen sind 1,2,6,22,90,394,1806,8558,...http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?infseq.


  • Schwache Primzahlen en

    Schwache Primzahlen sind Primzahlen, die nach Änderung einer beliebigen einzelnen Ziffer in eine beliebige andere Ziffer keine Primzahl mehr sind.


  • sechste Smarandache-Konstante Show Notations en

    Die sechste Smarandache-Konstante http://mathhub.info/smglom/smglom/smarandacheconstant6.omdoc?smarandacheconstant6?sixth-smarandache-constant ist definiert als

    http://mathhub.info/smglom/smglom/smarandacheconstant6.omdoc?smarandacheconstant6?sixth-smarandache-constant

    Dabei ist S(n)http://mathhub.info/smglom/smglom/smarandachefunction.omdoc?smarandachefunction?smarandache-functionn die Smarandache-Funktion.

    Die Summe konvergiert.

    (multi-relation-expression0.218282lethanlethan0.5)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/smglom/smarandacheconstant6.omdoc?smarandacheconstant6?sixth-smarandache-constanthttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethan

    .


  • sechzehnte Smarandache-Konstante en

    Die sechzehnte Smarandache-Konstante sixteenth-smarandache-constanthttp://mathhub.info/smglom/smglom/smarandacheconstant16.omdoc?smarandacheconstant16?sixteenth-smarandache-constant ist definiert als

    s16(α)http://mathhub.info/smglom/smglom/smarandacheconstant16.omdoc?smarandacheconstant16?sixteenth-smarandache-constantα

    Dabei ist S(n)http://mathhub.info/smglom/smglom/smarandachefunction.omdoc?smarandachefunction?smarandache-functionn die Smarandache-Funktion.

    Die Summe konvergiert für alle reellen Zahlen α>1http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanα.


  • Sehne en

    Wenn in einem Graph GG, der einen Kreis CC enthält, eine Kante von GG zwei Ecken des Kreises verbindet, selbst aber nicht zum Kreis gehört, so ist diese Kante eine Sehne des Kreises CC. Ein Kreis CC in GG ist genau dann sehnenlos, wenn er als Teilgraph in GG induziert ist.


  • sehnenlos en

    Wenn in einem Graph GG, der einen Kreis CC enthält, eine Kante von GG zwei Ecken des Kreises verbindet, selbst aber nicht zum Kreis gehört, so ist diese Kante eine Sehne des Kreises CC. Ein Kreis CC in GG ist genau dann sehnenlos, wenn er als Teilgraph in GG induziert ist.


  • Selbst-Primzahl en

    Eine Selbst-Primzahl ist eine Selbst-Zahl, die Primzahl ist.


  • Senke

    Sei G=(V,E)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalGhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE ein gerichteter Graph, dann nennen wir einen Knoten vVhttp://mathhub.info/smglom/sets/set.omdoc?set?insetvV

    • initial (oder eine Quelle) in GG, wenn es kein wVhttp://mathhub.info/smglom/sets/set.omdoc?set?insetwV gibt, so dass (w,v)Ehttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairwvE.

    • terminal (oder Senke) in GG, wenn es kein wVhttp://mathhub.info/smglom/sets/set.omdoc?set?insetwV gibt, so dass (v,w)Ehttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairvwE.


  • Sexy-Primzahlen en

    Sexy-Primzahlen sind Primzahlen, die um 6 differieren.


  • sichere Primzahl en

    Eine sichere Primzahl pp ist eine Primzahl, für die (p-1)2http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionp auch eine Primzahl ist.


  • siebente Smarandache-Konstante Show Notations en

    Die siebente Smarandache-Konstante http://mathhub.info/smglom/smglom/smarandacheconstant7.omdoc?smarandacheconstant7?seventh-smarandache-constant ist für eine natürliche Zahl kk definiert als

    http://mathhub.info/smglom/smglom/smarandacheconstant7.omdoc?smarandacheconstant7?seventh-smarandache-constant

    Dabei ist S(n)http://mathhub.info/smglom/smglom/smarandachefunction.omdoc?smarandachefunction?smarandache-functionn die Smarandache-Funktion.

    Die Summe konvergiert.


  • Sierpinski-Zahl en

  • Skewes-Zahl Show Notations en

    Die zweite Skewes-Zahl http://mathhub.info/smglom/smglom/skewesnumber.omdoc?skewesnumber?second-Skewes-number ist eine Obergrenze bis zu der nicht immer (π(n))(Li(n))http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?NumberPrimeNumbernhttp://mathhub.info/smglom/smglom/logarithmicintegralbig.omdoc?logarithmicintegralbig?logarithmicintbign gilt, vorausgesetzt, die Riemann-Hypothese ist falsch.

    =(10(10(101000)))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/smglom/skewesnumber.omdoc?skewesnumber?second-Skewes-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiation

  • small-set en

    Ein small-set positiver ganzer Zahlen

    S=({(s0,s1,s2,s3,)})http://mathhub.info/smglom/mv/equal.omdoc?equal?equalShttp://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?aseqdotsOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6ef

    ist eine unendliche Menge, für die die Summe

    (1s0)+(1s1)+(1s2)+(1s3)+http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionOMFOREIGNscala.xml.Node$@6040b6ef

    konvergiert.


  • Smarandache-Funktion en

    Die Smarandache-Funktion S(n)http://mathhub.info/smglom/smglom/smarandachefunction.omdoc?smarandachefunction?smarandache-functionn oder S(n)http://mathhub.info/smglom/smglom/smarandachefunction.omdoc?smarandachefunction?smarandache-functionn ist für eine gegebene positive ganze Zahl nn die kleinste natürliche Zahl, deren Fakultät durch nn teilbar ist.

    S(n)http://mathhub.info/smglom/smglom/smarandachefunction.omdoc?smarandachefunction?smarandache-functionn

  • Smarandache-Wellin-Primzahl

    Eine Smarandache-Wellin-Primzahl ist eine Smarandache-Wellin-Zahl, die auch Primzahl ist.


  • solitary

    Eine natürliche Zahl, die zu keinem freundlichem Paar gehört, wird als solitary bezeichnet.


  • Sophie-Germain-Primzahl en

    Eine Primzahl pp ist eine Sophie-Germain-Primzahl wenn (2p)+1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationp auch eine Primzahl ist.


  • Sphenische Zahlen en

  • stabile Mengen

    Paarweise nichtadjazente Ecken oder Kanten eines Graph GG werden unabhängig genannt. Eine Menge von Ecken oder Kanten wird unabhängig genannt, wenn es keine zwei Elemente gibt, die adjazent sind. Unabhängige Eckenmengen werden auch stabile Mengen genannt.


  • Stammbruch en

    Ein Stammbruch ist eine als Bruch geschriebene rationale Zahl, deren Zähler 1 und deren Nenner eine positive ganze Zahl ist.


  • starke Primzahl en

    Eine starke Primzahl ist eine Primzahl, die größer als das arithmetischen Mittel der nächstkleineren und der nächstgrößeren Primzahl ist.

    (pn)>(((p(n-1))+(p(n+1)))2)http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanhttp://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?prime-numbernhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?prime-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?prime-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionn

    Dabei ist pnhttp://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?prime-numbern die nn-te Primzahl.


  • Startknoten en

    Ist G=(V,E)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalGhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE gerichteter Graph, so nennen wir einen Vektor (p=((v0,,vn)))(V(n+1))http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalphttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlivnhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?nCartSpaceVhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionn einen Pfad in GG wenn (vi-1,vi)Ehttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6efE für alle 1inhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?betweeneein mit n>0http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethann.

    • v0OMFOREIGNscala.xml.Node$@6040b6ef heißt der Startknoten von pp (schreibe start(p)http://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pstartp )

    • vnOMFOREIGNscala.xml.Node$@6040b6ef heißt der Endknoten von pp (schreibe end(p)http://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pendp )

    • nn heißt die Länge von pp (schreibe len(p)http://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?plenp )


  • Startwert en

    Wir definieren das Produkt über eine Folge aiOMFOREIGNscala.xml.Node$@6040b6ef durch

    (i=nmai):=(defined-piecewise(
    0wenn(n=m)
    )
    (
    (an)sonst
    )
    )
    http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdfromtonmiOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnmhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionOMFOREIGNscala.xml.Node$@6040b6ef

    Die Variable ii wird der Laufindex oder Multiplikationsindex genannt, nn die untere (oder Startwert) und mm die obereGrenze (oder Endwert) des Produkts, zusammen bestimmen sie den Multiplikationsbereich.

    Gebräuchlich sind auch die fogenden Varianten des Produktoperators: φaihttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdCollφOMFOREIGNscala.xml.Node$@6040b6ef und iSaihttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdInCollSiOMFOREIGNscala.xml.Node$@6040b6ef. Der erste spezifiziert den Multiplikationsbereich durch eine Formel φφ über ii und der zweite gibt den Multiplikationsbereich direkt als eine Menge SS an.


  • Startwert en

    Summation ist iterierte Addition. Wir definieren die Summe über eine Folge aiOMFOREIGNscala.xml.Node$@6040b6ef durch

    (i=nmai):=(defined-piecewise(
    0wenn(n=m)
    )
    (
    (an)sonst
    )
    )
    http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/numberfields/sum.omdoc?sum?sumnmiOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnmhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionOMFOREIGNscala.xml.Node$@6040b6ef

    Die Variable ii wird der Laufindex oder Summationsindex genannt, nn die untere (oder Startwert) und mm die obereGrenze (oder Endwert) der Summe, zusammen bestimmen sie den Summationsbereich.

    Gebräuchlich sind auch die fogenden Varianten des Summenoperators: (SumCollφ[i]ai)http://mathhub.info/smglom/numberfields/sum.omdoc?sum?SumCollφiOMFOREIGNscala.xml.Node$@6040b6ef und iSaihttp://mathhub.info/smglom/numberfields/sum.omdoc?sum?SumInCollSiOMFOREIGNscala.xml.Node$@6040b6ef. Der erste spezifiziert den Summationsbereich durch eine Formel φφ über ii und der zweite gibt den Summationsbereich als eine Menge SS an.


  • Stern-Primzahl en

    Wenn es für eine Primzahl qq keine kleinere Primzahl pp und eine positive ganze Zahl bb gibt, so dass

    q=(p+(2(b2)))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalqhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationb

    gilt, dann ist qq eine Stern-Primzahl.


  • stetig en

    Sind X,Ohttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureXOMFOREIGNscala.xml.Node$@6040b6ef und X,Ohttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6ef topologische Räume, so nennen wir eine Funktion f:XYhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfXY stetig, wenn ((𝐈𝐦(f))X)Ohttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/sets/image.omdoc?image?imagefXOMFOREIGNscala.xml.Node$@6040b6ef für alle XOhttp://mathhub.info/smglom/sets/set.omdoc?set?insetXOMFOREIGNscala.xml.Node$@6040b6ef.


  • Stoneham-Zahlen en

    Die Stoneham-Zahlen αb,chttp://mathhub.info/smglom/smglom/stonehamnumber.omdoc?stonehamnumber?stonehambc werden für teilerfremde Zahlen (morethanbc1)http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanbc definiert als

    αb,chttp://mathhub.info/smglom/smglom/stonehamnumber.omdoc?stonehamnumber?stonehambc

  • strikte Ordnung en

    Eine quasiordnung pole(A×A)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsethttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?polehttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAA auf AA heißt partielle Ordnung, wenn sie antisymmetrisch ist. polehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole induziert eine strikte Ordnung poless:=(bsetst[ab](a,b)pole)http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?polesshttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstabhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairabhttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole.

    Die konversen Relationen schreiben wir oft als pomehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pome und pomorehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pomore.

    Wir nennen eine Struktur S,polehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureShttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole aus einer Menge SS und einer partiellen Ordnung polehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole eine geordnete Menge.

    Eine partielle Ordnung RR heißt totale Ordnung (or einfache Ordnung oder lineare Ordnung), falls abhttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?poleab oder bahttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?poleba für alle a,bAhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabA.

    Wir nennen eine Struktur S,polehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureShttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole aus einer Menge SS und einer totalen Ordnung polehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole eine total geordnete Menge.


  • Struktur Show Notations en

    Eine Struktur fasst mehrere existierende mathematische Objekte (die Komponenten) zu einem neuen Objekt zusammen. Strukturen werden normalerweise als endliche Aufzählungen ihrer Komponenten gegeben, die durch spezielle Namen referenziert werden können.


  • stückweise definiert Show Notations en

    Eine Abbildung mm ist stückweise definiert, schreibe

    (mx)=(defined-piecewise(
    a1wennA1
    )
    (
    wenn
    )
    (
    anwennAn
    )
    (
    osonst
    )
    )
    http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationmxhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?pieceOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?pieceOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwiseo

    wobei AiOMFOREIGNscala.xml.Node$@6040b6ef Bedingungen an xx sind, wenn (mx)=aihttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationmxOMFOREIGNscala.xml.Node$@6040b6ef für alle xx so dass AiOMFOREIGNscala.xml.Node$@6040b6ef und oo sonst.


  • subharmonische Reihe en

    Sei Mhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetMhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number, dann nennen wir eine unendliche Reihe nM(1n)http://mathhub.info/smglom/numberfields/sum.omdoc?sum?SumInCollMnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionn eine subharmonische Reihe.


  • Subtraktion Show Notations

    Die arithmetischen Operationen sind Addition additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?addition, Subtraktion, subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction, Multiplikation multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplication, Division divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?division und Exponentiation.


  • sum of squares Show Notations en

    Die Funktion rn(k)http://mathhub.info/smglom/smglom/sumofsquaresfunction.omdoc?sumofsquaresfunction?sumof-squares-functionnk wird im Englischen als sum of squares bezeichnet.

    (rn(k))=(#((bsetst[an](a1,a2,,an)(n))))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/smglom/sumofsquaresfunction.omdoc?sumofsquaresfunction?sumof-squares-functionnkhttp://mathhub.info/smglom/sets/finite-cardinality.omdoc?finite-cardinality?cardinalityhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstanhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tupleOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?nCartSpacehttp://mathhub.info/smglom/numberfields/integernumbers.omdoc?integernumbers?integersn

  • Summation

    Summation ist iterierte Addition. Wir definieren die Summe über eine Folge aiOMFOREIGNscala.xml.Node$@6040b6ef durch

    (i=nmai):=(defined-piecewise(
    0wenn(n=m)
    )
    (
    (an)sonst
    )
    )
    http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/numberfields/sum.omdoc?sum?sumnmiOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnmhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionOMFOREIGNscala.xml.Node$@6040b6ef

    Die Variable ii wird der Laufindex oder Summationsindex genannt, nn die untere (oder Startwert) und mm die obereGrenze (oder Endwert) der Summe, zusammen bestimmen sie den Summationsbereich.

    Gebräuchlich sind auch die fogenden Varianten des Summenoperators: (SumCollφ[i]ai)http://mathhub.info/smglom/numberfields/sum.omdoc?sum?SumCollφiOMFOREIGNscala.xml.Node$@6040b6ef und iSaihttp://mathhub.info/smglom/numberfields/sum.omdoc?sum?SumInCollSiOMFOREIGNscala.xml.Node$@6040b6ef. Der erste spezifiziert den Summationsbereich durch eine Formel φφ über ii und der zweite gibt den Summationsbereich als eine Menge SS an.


  • Summationsbereich en

    Summation ist iterierte Addition. Wir definieren die Summe über eine Folge aiOMFOREIGNscala.xml.Node$@6040b6ef durch

    (i=nmai):=(defined-piecewise(
    0wenn(n=m)
    )
    (
    (an)sonst
    )
    )
    http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/numberfields/sum.omdoc?sum?sumnmiOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnmhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionOMFOREIGNscala.xml.Node$@6040b6ef

    Die Variable ii wird der Laufindex oder Summationsindex genannt, nn die untere (oder Startwert) und mm die obereGrenze (oder Endwert) der Summe, zusammen bestimmen sie den Summationsbereich.

    Gebräuchlich sind auch die fogenden Varianten des Summenoperators: (SumCollφ[i]ai)http://mathhub.info/smglom/numberfields/sum.omdoc?sum?SumCollφiOMFOREIGNscala.xml.Node$@6040b6ef und iSaihttp://mathhub.info/smglom/numberfields/sum.omdoc?sum?SumInCollSiOMFOREIGNscala.xml.Node$@6040b6ef. Der erste spezifiziert den Summationsbereich durch eine Formel φφ über ii und der zweite gibt den Summationsbereich als eine Menge SS an.


  • Summationsindex en

    Summation ist iterierte Addition. Wir definieren die Summe über eine Folge aiOMFOREIGNscala.xml.Node$@6040b6ef durch

    (i=nmai):=(defined-piecewise(
    0wenn(n=m)
    )
    (
    (an)sonst
    )
    )
    http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/numberfields/sum.omdoc?sum?sumnmiOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnmhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionOMFOREIGNscala.xml.Node$@6040b6ef

    Die Variable ii wird der Laufindex oder Summationsindex genannt, nn die untere (oder Startwert) und mm die obereGrenze (oder Endwert) der Summe, zusammen bestimmen sie den Summationsbereich.

    Gebräuchlich sind auch die fogenden Varianten des Summenoperators: (SumCollφ[i]ai)http://mathhub.info/smglom/numberfields/sum.omdoc?sum?SumCollφiOMFOREIGNscala.xml.Node$@6040b6ef und iSaihttp://mathhub.info/smglom/numberfields/sum.omdoc?sum?SumInCollSiOMFOREIGNscala.xml.Node$@6040b6ef. Der erste spezifiziert den Summationsbereich durch eine Formel φφ über ii und der zweite gibt den Summationsbereich als eine Menge SS an.


  • Summe Show Notations en

    Summation ist iterierte Addition. Wir definieren die Summe über eine Folge aiOMFOREIGNscala.xml.Node$@6040b6ef durch

    (i=nmai):=(defined-piecewise(
    0wenn(n=m)
    )
    (
    (an)sonst
    )
    )
    http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/numberfields/sum.omdoc?sum?sumnmiOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnmhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionOMFOREIGNscala.xml.Node$@6040b6ef

    Die Variable ii wird der Laufindex oder Summationsindex genannt, nn die untere (oder Startwert) und mm die obereGrenze (oder Endwert) der Summe, zusammen bestimmen sie den Summationsbereich.

    Gebräuchlich sind auch die fogenden Varianten des Summenoperators: (SumCollφ[i]ai)http://mathhub.info/smglom/numberfields/sum.omdoc?sum?SumCollφiOMFOREIGNscala.xml.Node$@6040b6ef und iSaihttp://mathhub.info/smglom/numberfields/sum.omdoc?sum?SumInCollSiOMFOREIGNscala.xml.Node$@6040b6ef. Der erste spezifiziert den Summationsbereich durch eine Formel φφ über ii und der zweite gibt den Summationsbereich als eine Menge SS an.


  • summierte Mangoldt-Funktion en

    Die summierte Mangoldt-Funktion

    ψψ

    wird auch als Tschebyschow-Funktion bezeichnet.


  • super-Catalan-Zahl Show Notations en

    Die Schröder-Hipparchus-Zahl (oder super-Catalan-Zahl) S(n)http://mathhub.info/smglom/numbers/schroederhipparchusnumberformula.omdoc?schroederhipparchusnumberformula?Schroeder-Hipparchus-numbern ist für natürliche Zahl nn definiert als

    S(n)http://mathhub.info/smglom/numbers/schroederhipparchusnumberformula.omdoc?schroederhipparchusnumberformula?Schroeder-Hipparchus-numbern

  • super-Catalan-Zahlen ) en

    Die Schröder-Hipparchus-Zahlen (oder super-Catalan-Zahlen ) S(n)http://mathhub.info/smglom/numbers/schroederhipparchusnumberrec.omdoc?schroederhipparchusnumberrec?schroederhipprecn werden für natürliche Zahl nn rekursiv definiert:

    S(1)http://mathhub.info/smglom/numbers/schroederhipparchusnumberrec.omdoc?schroederhipparchusnumberrec?schroederhipprec equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal (S(2))=1http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/schroederhipparchusnumberrec.omdoc?schroederhipparchusnumberrec?schroederhipprec
    S(n)http://mathhub.info/smglom/numbers/schroederhipparchusnumberrec.omdoc?schroederhipparchusnumberrec?schroederhipprecn equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal (1n)((((6n)-9)(S((n-1))))-((n-3)(S((n-2)))))http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnhttp://mathhub.info/smglom/numbers/schroederhipparchusnumberrec.omdoc?schroederhipparchusnumberrec?schroederhipprechttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numbers/schroederhipparchusnumberrec.omdoc?schroederhipparchusnumberrec?schroederhipprechttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn

  • supremum Show Notations

    Sei S,polehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureShttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole eine geordnete Menge und TShttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetTS, dann nennen wir die kleinste obere Schranke sup(T)http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?supremumT (größte untere Schranke inf(T)http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?infimumT) von TT das Supremum (Infimum) von TT (falls dies existiert).

    Ist ee ein Ausdruck und φφ eine Bedingung (in einer Variablen xx), so schreiben wir (bsupremumφ[x]e)http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?bsupremumφxe für sup((bsetst[x]φ))http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?supremumhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstxφ und nennen es das supremum für ee über φφ. Analog schreiben wir (binfimumφ[x]e)http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?binfimumφxe für inf((bsetst[x]φ))http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?infimumhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstxφ und nennen es das infimum für ee über φφ.


  • Supremum Show Notations en

    Sei S,polehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureShttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole eine geordnete Menge und TShttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetTS, dann nennen wir die kleinste obere Schranke sup(T)http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?supremumT (größte untere Schranke inf(T)http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?infimumT) von TT das Supremum (Infimum) von TT (falls dies existiert).

    Ist ee ein Ausdruck und φφ eine Bedingung (in einer Variablen xx), so schreiben wir (bsupremumφ[x]e)http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?bsupremumφxe für sup((bsetst[x]φ))http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?supremumhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstxφ und nennen es das supremum für ee über φφ. Analog schreiben wir (binfimumφ[x]e)http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?binfimumφxe für inf((bsetst[x]φ))http://mathhub.info/smglom/calculus/supinf.omdoc?supinf?infimumhttp://mathhub.info/smglom/sets/set.omdoc?set?bsetstxφ und nennen es das infimum für ee über φφ.


  • surjektiv en

    Eine Funktion f:SThttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfST heißt surjektiv, wenn es für alle yThttp://mathhub.info/smglom/sets/set.omdoc?set?insetyT ein xShttp://mathhub.info/smglom/sets/set.omdoc?set?insetxS gibt mit (f)=yhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalfy.


  • Symmetrie en

    Für eine Mente MM nennen wir eine Funktion d:(M×M)http://mathhub.info/smglom/sets/functions.omdoc?functions?fundhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsMMhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number eine Abstandsfunktion (or Metrik ) auf MM, falls die folgenden drei Eigenschaften für alle x,y,zMhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetxyzM gelten:

    1. 1.

      (d)=0http://mathhub.info/smglom/mv/equal.omdoc?equal?equald gdw. x=yhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalxy (Definitheit),

    2. 2.

      (d)=(d)http://mathhub.info/smglom/mv/equal.omdoc?equal?equaldd (Symmetrie) und

    3. 3.

      (d)((d)+(d))http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethandhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additiondd (Dreiecksungleichung).

    Wir nennen M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd einen metrischen Raum mit Grundmenge MM und Metrik dd.


  • symmetrisch en

    Eine Relation R(A×A)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetRhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAA heißt

    • symmetrisch auf AA, falls (b,a)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairbaR für alle a,bAhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabA mit (a,b)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairabR.

    • asymmetrisch auf AA, falls (b,a)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?ninsethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairbaR für alle a,bAhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabA mit (a,b)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairabR.

    • antisymmetrisch auf AA, falls a=bhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalab wenn (a,b)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairabR und (b,a)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairbaR.


  • Taillenweite Show Notations en

    Für einen Graph GG ist die minimale Länge eines in GG enthaltenen Kreises die Taillenweite g(G)http://mathhub.info/smglom/graphs/graphcycleparameters.omdoc?graphcycleparameters?girthG von GG, die maximale Läenge eines in GG enthaltenen Kreises ist der Umfang von GG. Für einen Graph, der keinen Kreis enthält, setzen wir die Taillenweite auf (g(G))=http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/graphs/graphcycleparameters.omdoc?graphcycleparameters?girthGhttp://mathhub.info/smglom/numberfields/infinity.omdoc?infinity?infinity, sein Umfang wird auf Null gesetzt.


  • Taxicab-Zahl Show Notations en

    Die nn-te Taxicab-Zahl Ta(n)http://mathhub.info/smglom/numbers/taxicabnumber.omdoc?taxicabnumber?taxicab-numbern ist definiert als die kleinste natürliche Zahl, die sich auf nn verschiedene Arten als Summe zweier Kubikzahlen darstellen lässt.

    Zum Beispiel:

    (multi-relation-expressionTa(3)equal87539319equal(1673)+(4363))http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionhttp://mathhub.info/smglom/numbers/taxicabnumber.omdoc?taxicabnumber?taxicab-numberhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiation
    equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal
    equalhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equal

  • Taylorreihe Show Notations en

    Sei ff eine reellwertige oder komplexe Funktion die glatt ist auf einem Häufungspunkt aa des Definitionsbereichs von ff, dann nennen wir die Reihe

    (fundefeq[fxa]𝑇f(x;a))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfxahttp://mathhub.info/smglom/calculus/Taylor-series.omdoc?Taylor-series?Taylor-seriesfxa

    die Taylorreihe für ff am Entwicklungspunkt aa. Ist a=0http://mathhub.info/smglom/mv/equal.omdoc?equal?equala, so nennen wir die Reihe die Mclaurinreihe.


  • te Ableitung en

    Wir nennen ff nn mal differeinzierbar in aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM, falls dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa als Grenzwert existiert. Analog nennen wir ff nn mal differenzierbar auf MM, falls dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx existiert und auf MM total ist.

    Wir nennen ff unendlich differenzierbar oder glatt auf aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM oder MM, wenn dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa beziehungsweise dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx für alle nhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number existieren.

    Für ein nhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number definieren wir die nnte Ableitung einer Funktion f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN durch

    (fundefeq[nfxa]dnfdxn|x=a)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqnfxahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa

    Die erste Ableitung d1dx1fhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtfx von ff ist Dxfhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfx, d2dx2fhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtfx ist die zweite Ableitung von ff, d3dx3fhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtfx die dritte Ableitung von ff, usw. In der Leibniz Notation wird die nnte Ableiguntsfunktion von ff als dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx geschrieben.


  • te Ableitung en

    Wir nennen ff nn mal differeinzierbar in aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM, falls dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa als Grenzwert existiert. Analog nennen wir ff nn mal differenzierbar auf MM, falls dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx existiert und auf MM total ist.

    Wir nennen ff unendlich differenzierbar oder glatt auf aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM oder MM, wenn dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa beziehungsweise dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx für alle nhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number existieren.

    Für ein nhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number definieren wir die nnte Ableitung einer Funktion f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN durch

    (fundefeq[nfxa]dnfdxn|x=a)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqnfxahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa

    Die erste Ableitung d1dx1fhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtfx von ff ist Dxfhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfx, d2dx2fhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtfx ist die zweite Ableitung von ff, d3dx3fhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtfx die dritte Ableitung von ff, usw. In der Leibniz Notation wird die nnte Ableiguntsfunktion von ff als dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx geschrieben.


  • teilbar Show Notations en

    Sei Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR ein Ring und a,bRhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabR mit a0http://mathhub.info/smglom/mv/equal.omdoc?equal?notequala, dann nennen wir aa einen linken Teiler von bb (und sagen aa teilt bb links oder bb ist link teilbar durch aa) und bb ein linkes Vielfaches von aa, wenn es ein nRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnR gibt, für die (multiplication)=bhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationb ist.

    Analog nennen wir aa einen rechten Teiler von bb (und sagen aa teilt bb rechts oder bb ist rechts teilbar durch aa) und bb ein rechtes Vielfaches von aa, wenn es ein nRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnR gibt, für die (na)bhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnab ist.

    Ist RR kommutativ, so fallen rechte und linke Teiler zusammen, und wir sprechen einfach von Teiler, teilbar, und Vielfachen. Wir schreiben dann a|bhttp://mathhub.info/smglom/algebra/ring-divisor.omdoc?ring-divisor?divisorab.

    Wir nennen 1, 1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction, nn und nhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn triviale divisoren von nn. Ein Teiler von nn, der nicht trivial ist, heißt echter Teiler von nn.


  • Teiler Show Notations en

    Sei Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR ein Ring und a,bRhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabR mit a0http://mathhub.info/smglom/mv/equal.omdoc?equal?notequala, dann nennen wir aa einen linken Teiler von bb (und sagen aa teilt bb links oder bb ist link teilbar durch aa) und bb ein linkes Vielfaches von aa, wenn es ein nRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnR gibt, für die (multiplication)=bhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationb ist.

    Analog nennen wir aa einen rechten Teiler von bb (und sagen aa teilt bb rechts oder bb ist rechts teilbar durch aa) und bb ein rechtes Vielfaches von aa, wenn es ein nRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnR gibt, für die (na)bhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnab ist.

    Ist RR kommutativ, so fallen rechte und linke Teiler zusammen, und wir sprechen einfach von Teiler, teilbar, und Vielfachen. Wir schreiben dann a|bhttp://mathhub.info/smglom/algebra/ring-divisor.omdoc?ring-divisor?divisorab.

    Wir nennen 1, 1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction, nn und nhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn triviale divisoren von nn. Ein Teiler von nn, der nicht trivial ist, heißt echter Teiler von nn.


  • Teileranzahlfunktion Show Notations

    Die Teileranzahlfunktion dnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationdn ist die Summe der positiven Teiler von nn.

    dd

    Andere Symbole: τ(n)http://mathhub.info/smglom/smglom/numberofdivisorsfunction.omdoc?numberofdivisorsfunction?numberofdivisorsfunctionn, τ(n)http://mathhub.info/smglom/smglom/numberofdivisorsfunction.omdoc?numberofdivisorsfunction?numberofdivisorsfunctionn, τ(n)http://mathhub.info/smglom/smglom/numberofdivisorsfunction.omdoc?numberofdivisorsfunction?numberofdivisorsfunctionn


  • teilerfremd Show Notations en

    Zwei natürliche Zahlen sind genau dann teilerfremd oder relativ prim, wenn deren größter gemeinsamer Teiler 1 ist.


  • Teilersummenfunktion Show Notations en

    Die Teilersummenfunktion σ1(n)http://mathhub.info/smglom/smglom/sumofdivisorsfunction.omdoc?sumofdivisorsfunction?sumdiv-functionn (auch σ1(n)http://mathhub.info/smglom/smglom/sumofdivisorsfunction.omdoc?sumofdivisorsfunction?sumdiv-functionn) ist die Summe der positiven Teiler von nn, also σ1(n)http://mathhub.info/smglom/smglom/sumofdivisorsfunction.omdoc?sumofdivisorsfunction?sumdiv-functionn.


  • Teilmenge Show Notations ro tr en

    Eine Menge AA ist eine Teilmenge einer Menge BB (schreibe ABhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?proper-subsetAB), wenn alle xAhttp://mathhub.info/smglom/sets/set.omdoc?set?insetxA Elemente von BB sind.


  • teilt links en

    Sei Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR ein Ring und a,bRhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabR mit a0http://mathhub.info/smglom/mv/equal.omdoc?equal?notequala, dann nennen wir aa einen linken Teiler von bb (und sagen aa teilt bb links oder bb ist link teilbar durch aa) und bb ein linkes Vielfaches von aa, wenn es ein nRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnR gibt, für die (multiplication)=bhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationb ist.

    Analog nennen wir aa einen rechten Teiler von bb (und sagen aa teilt bb rechts oder bb ist rechts teilbar durch aa) und bb ein rechtes Vielfaches von aa, wenn es ein nRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnR gibt, für die (na)bhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnab ist.

    Ist RR kommutativ, so fallen rechte und linke Teiler zusammen, und wir sprechen einfach von Teiler, teilbar, und Vielfachen. Wir schreiben dann a|bhttp://mathhub.info/smglom/algebra/ring-divisor.omdoc?ring-divisor?divisorab.

    Wir nennen 1, 1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction, nn und nhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn triviale divisoren von nn. Ein Teiler von nn, der nicht trivial ist, heißt echter Teiler von nn.


  • teilt rechts en

    Sei Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR ein Ring und a,bRhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabR mit a0http://mathhub.info/smglom/mv/equal.omdoc?equal?notequala, dann nennen wir aa einen linken Teiler von bb (und sagen aa teilt bb links oder bb ist link teilbar durch aa) und bb ein linkes Vielfaches von aa, wenn es ein nRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnR gibt, für die (multiplication)=bhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationb ist.

    Analog nennen wir aa einen rechten Teiler von bb (und sagen aa teilt bb rechts oder bb ist rechts teilbar durch aa) und bb ein rechtes Vielfaches von aa, wenn es ein nRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnR gibt, für die (na)bhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnab ist.

    Ist RR kommutativ, so fallen rechte und linke Teiler zusammen, und wir sprechen einfach von Teiler, teilbar, und Vielfachen. Wir schreiben dann a|bhttp://mathhub.info/smglom/algebra/ring-divisor.omdoc?ring-divisor?divisorab.

    Wir nennen 1, 1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction, nn und nhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn triviale divisoren von nn. Ein Teiler von nn, der nicht trivial ist, heißt echter Teiler von nn.


  • Teilüberdeckung en

    Eine Überdeckung einer Menge XX ist eine Familie CC von Mengen mit X(aAUa)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetXhttp://mathhub.info/smglom/sets/union.omdoc?union?munionCollectionAaOMFOREIGNscala.xml.Node$@6040b6ef. Eine Teilmenge CChttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efC heißt Teilüberdeckung von XX, wenn sie XX immer noch überdeckt.


  • terminal

    Sei G=(V,E)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalGhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE ein gerichteter Graph, dann nennen wir einen Knoten vVhttp://mathhub.info/smglom/sets/set.omdoc?set?insetvV

    • initial (oder eine Quelle) in GG, wenn es kein wVhttp://mathhub.info/smglom/sets/set.omdoc?set?insetwV gibt, so dass (w,v)Ehttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairwvE.

    • terminal (oder Senke) in GG, wenn es kein wVhttp://mathhub.info/smglom/sets/set.omdoc?set?insetwV gibt, so dass (v,w)Ehttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairvwE.


  • Thabit-Zahl en

    Eine Thabit-Zahl ist eine ganze Zahl der Form

    (32n)-1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn

    mit der nicht-negativen ganzen Zahl nn.


  • Topologie en

    Ein topologischer Raum X,Ohttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureXOMFOREIGNscala.xml.Node$@6040b6ef ist eine Menge XX zusammen mit einer Familie O(𝒫(X))http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/sets/powerset.omdoc?powerset?powersetX, so daß

    1. 1.

      ,XOhttp://mathhub.info/smglom/sets/set.omdoc?set?minsethttp://mathhub.info/smglom/sets/emptyset.omdoc?emptyset?esetXOMFOREIGNscala.xml.Node$@6040b6ef.

    2. 2.

      (SSs)Ohttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/union.omdoc?union?munionCollectionOMFOREIGNscala.xml.Node$@6040b6efSsOMFOREIGNscala.xml.Node$@6040b6ef falls SShttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetSOMFOREIGNscala.xml.Node$@6040b6ef.

    3. 3.

      (SSs)Ohttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/intersection.omdoc?intersection?mintersectCollectionOMFOREIGNscala.xml.Node$@6040b6efSsOMFOREIGNscala.xml.Node$@6040b6ef falls SShttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetSOMFOREIGNscala.xml.Node$@6040b6ef endlich.

    Dann heißt OOMFOREIGNscala.xml.Node$@6040b6ef eine Topologie) auf XX. Elemente der Topologie OOMFOREIGNscala.xml.Node$@6040b6ef heißen offene Mengen und ihre Komplemente abgeschlossen oder einfach geschlossen . Eine Teilmenge von XX kann weder gesclossen noch offen, oder gesclossen, oder offen oder beides.


  • topologischer Raum en

    Ein topologischer Raum X,Ohttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureXOMFOREIGNscala.xml.Node$@6040b6ef ist eine Menge XX zusammen mit einer Familie O(𝒫(X))http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/sets/powerset.omdoc?powerset?powersetX, so daß

    1. 1.

      ,XOhttp://mathhub.info/smglom/sets/set.omdoc?set?minsethttp://mathhub.info/smglom/sets/emptyset.omdoc?emptyset?esetXOMFOREIGNscala.xml.Node$@6040b6ef.

    2. 2.

      (SSs)Ohttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/union.omdoc?union?munionCollectionOMFOREIGNscala.xml.Node$@6040b6efSsOMFOREIGNscala.xml.Node$@6040b6ef falls SShttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetSOMFOREIGNscala.xml.Node$@6040b6ef.

    3. 3.

      (SSs)Ohttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/intersection.omdoc?intersection?mintersectCollectionOMFOREIGNscala.xml.Node$@6040b6efSsOMFOREIGNscala.xml.Node$@6040b6ef falls SShttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetSOMFOREIGNscala.xml.Node$@6040b6ef endlich.

    Dann heißt OOMFOREIGNscala.xml.Node$@6040b6ef eine Topologie) auf XX. Elemente der Topologie OOMFOREIGNscala.xml.Node$@6040b6ef heißen offene Mengen und ihre Komplemente abgeschlossen oder einfach geschlossen . Eine Teilmenge von XX kann weder gesclossen noch offen, oder gesclossen, oder offen oder beides.


  • total

    Eine Relation R(A×B)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetRhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAB heißt total wenn es für alle xAhttp://mathhub.info/smglom/sets/set.omdoc?set?insetxA ein yBhttp://mathhub.info/smglom/sets/set.omdoc?set?insetyB gibt, so dass (x,y)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairxyR.


  • total geordnete Menge en

    Eine partielle Ordnung RR heißt totale Ordnung (or einfache Ordnung oder lineare Ordnung), falls abhttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?poleab oder bahttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?poleba für alle a,bAhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabA.

    Wir nennen eine Struktur S,polehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureShttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole aus einer Menge SS und einer totalen Ordnung polehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole eine total geordnete Menge.


  • totale Funktion en

    Ist f:XYhttp://mathhub.info/smglom/sets/functions.omdoc?functions?partfunfXY eine totale Relation (d.h. für alle xXhttp://mathhub.info/smglom/sets/set.omdoc?set?insetxX gibt es ein eindeutiges yYhttp://mathhub.info/smglom/sets/set.omdoc?set?insetyY it (x,y)fhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairxyf), dann nennen wir ff eine totale Funktion und schreiben f:XYhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfXY.


  • totale Ordnung en

    Eine partielle Ordnung RR heißt totale Ordnung (or einfache Ordnung oder lineare Ordnung), falls abhttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?poleab oder bahttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?poleba für alle a,bAhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabA.

    Wir nennen eine Struktur S,polehttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureShttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole aus einer Menge SS und einer totalen Ordnung polehttp://mathhub.info/smglom/sets/partial-order.omdoc?partial-order?pole eine total geordnete Menge.


  • transitiv en

    Eine Relation R(A×A)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetRhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAA heißt transitiv auf AA, falls (a,c)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairacR für alle a,b,cAhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabcA mit (a,b)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairabR und (b,c)Rhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairbcR.


  • transtiv-reflexiven Abschluss Show Notations en

    Ist RR eine binäre Relation, so nennen wir die kleinste transitive, reflexive Relation die RR enthält den transtiv-reflexiven Abschluss von RR und schreiben sie als Rhttp://mathhub.info/smglom/sets/transitive-closure.omdoc?transitive-closure?transitive-reflexive-closureR.


  • triviale divisor en

    Wir nennen 1, 1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction, nn und nhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn triviale divisoren von nn. Ein Teiler von nn, der nicht trivial ist, heißt echter Teiler von nn.


  • Tschebyschow-Funktion en

    Die summierte Mangoldt-Funktion

    ψψ

    wird auch als Tschebyschow-Funktion bezeichnet.


  • Umfang en

    Für einen Graph GG ist die minimale Länge eines in GG enthaltenen Kreises die Taillenweite g(G)http://mathhub.info/smglom/graphs/graphcycleparameters.omdoc?graphcycleparameters?girthG von GG, die maximale Läenge eines in GG enthaltenen Kreises ist der Umfang von GG. Für einen Graph, der keinen Kreis enthält, setzen wir die Taillenweite auf (g(G))=http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/graphs/graphcycleparameters.omdoc?graphcycleparameters?girthGhttp://mathhub.info/smglom/numberfields/infinity.omdoc?infinity?infinity, sein Umfang wird auf Null gesetzt.


  • Umgebung en

    Ist X,Ohttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureXOMFOREIGNscala.xml.Node$@6040b6ef ein topologischer Raum und pXhttp://mathhub.info/smglom/sets/set.omdoc?set?insetpX, so nennen wir ein offene Menge NOhttp://mathhub.info/smglom/sets/set.omdoc?set?insetNOMFOREIGNscala.xml.Node$@6040b6ef eine Umgebung von pp, falls pNhttp://mathhub.info/smglom/sets/set.omdoc?set?insetpN.


  • Umkehrung en

    Die Umkehrung einer Zahl abchttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationabc ist cbahttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationcba.


  • unabhängig

    Paarweise nichtadjazente Ecken oder Kanten eines Graph GG werden unabhängig genannt. Eine Menge von Ecken oder Kanten wird unabhängig genannt, wenn es keine zwei Elemente gibt, die adjazent sind. Unabhängige Eckenmengen werden auch stabile Mengen genannt.


  • undefiniert auf en

    Eine partielle Funktion f:XYhttp://mathhub.info/smglom/sets/functions.omdoc?functions?partfunfXY heißt undefiniert auf xXhttp://mathhub.info/smglom/sets/set.omdoc?set?insetxX (schreibe (f)=http://mathhub.info/smglom/mv/equal.omdoc?equal?equalfhttp://mathhub.info/smglom/sets/functions.omdoc?functions?undefd), wenn (x,y)fhttp://mathhub.info/smglom/sets/set.omdoc?set?ninsethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairxyf für alle yYhttp://mathhub.info/smglom/sets/set.omdoc?set?insetyY.


  • undefiniert auf en

    Eine partielle Funktion f:XYhttp://mathhub.info/smglom/sets/functions.omdoc?functions?partfunfXY heißt undefiniert auf xXhttp://mathhub.info/smglom/sets/set.omdoc?set?insetxX (schreibe (f)=http://mathhub.info/smglom/mv/equal.omdoc?equal?equalfhttp://mathhub.info/smglom/sets/functions.omdoc?functions?undefd), wenn (x,y)fhttp://mathhub.info/smglom/sets/set.omdoc?set?ninsethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairxyf für alle yYhttp://mathhub.info/smglom/sets/set.omdoc?set?insetyY.


  • unendlich en

    Eine Menge die nicht endlich ist heißt unendlich.


  • unendlich differenzierbar en

    Wir nennen ff nn mal differeinzierbar in aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM, falls dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa als Grenzwert existiert. Analog nennen wir ff nn mal differenzierbar auf MM, falls dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx existiert und auf MM total ist.

    Wir nennen ff unendlich differenzierbar oder glatt auf aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM oder MM, wenn dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa beziehungsweise dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx für alle nhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number existieren.


  • unendliche Summe Show Notations en

    Ist (sequenceon[a]n)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?sequenceonan eine Folge mit (an)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselanhttp://mathhub.info/smglom/numberfields/realnumbers.omdoc?realnumbers?real-number oder (an)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselanhttp://mathhub.info/smglom/numberfields/complexnumbers.omdoc?complexnumbers?complex-number, so definieren wir die unendliche Summe (infinite-sum1[n]an)http://mathhub.info/smglom/calculus/infinitesum.omdoc?infinitesum?infinite-sumnhttp://mathhub.info/smglom/calculus/sequences.omdoc?sequences?seqselan als den Grenzwert limn(sn)http://mathhub.info/smglom/calculus/sequenceConvergence.omdoc?sequenceConvergence?limitnhttp://mathhub.info/smglom/calculus/series.omdoc?series?partial-suman der Partialsummenfolge, falls dieser existiert.


  • Unendlichkeit Show Notations en

    Die Unendlichkeit (schreibe http://mathhub.info/smglom/numberfields/infinity.omdoc?infinity?infinity) ist ein abstraktes Konzept, das auf Begriffe und Objekte angewendet wird, die keine Grenze haben. In der Mathematik wird sie meist wie eine Zahl behandelt.


  • untere en

    Wir definieren das Produkt über eine Folge aiOMFOREIGNscala.xml.Node$@6040b6ef durch

    (i=nmai):=(defined-piecewise(
    0wenn(n=m)
    )
    (
    (an)sonst
    )
    )
    http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdfromtonmiOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnmhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionOMFOREIGNscala.xml.Node$@6040b6ef

    Die Variable ii wird der Laufindex oder Multiplikationsindex genannt, nn die untere (oder Startwert) und mm die obereGrenze (oder Endwert) des Produkts, zusammen bestimmen sie den Multiplikationsbereich.

    Gebräuchlich sind auch die fogenden Varianten des Produktoperators: φaihttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdCollφOMFOREIGNscala.xml.Node$@6040b6ef und iSaihttp://mathhub.info/smglom/numberfields/product.omdoc?product?ProdInCollSiOMFOREIGNscala.xml.Node$@6040b6ef. Der erste spezifiziert den Multiplikationsbereich durch eine Formel φφ über ii und der zweite gibt den Multiplikationsbereich direkt als eine Menge SS an.


  • untere en

    Summation ist iterierte Addition. Wir definieren die Summe über eine Folge aiOMFOREIGNscala.xml.Node$@6040b6ef durch

    (i=nmai):=(defined-piecewise(
    0wenn(n=m)
    )
    (
    (an)sonst
    )
    )
    http://mathhub.info/smglom/mv/defeq.omdoc?defeq?definitional-equationhttp://mathhub.info/smglom/numberfields/sum.omdoc?sum?sumnmiOMFOREIGNscala.xml.Node$@6040b6efhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?defined-piecewisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?piecehttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalnmhttp://mathhub.info/smglom/mv/piecewise.omdoc?piecewise?otherwisehttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionOMFOREIGNscala.xml.Node$@6040b6ef

    Die Variable ii wird der Laufindex oder Summationsindex genannt, nn die untere (oder Startwert) und mm die obereGrenze (oder Endwert) der Summe, zusammen bestimmen sie den Summationsbereich.

    Gebräuchlich sind auch die fogenden Varianten des Summenoperators: (SumCollφ[i]ai)http://mathhub.info/smglom/numberfields/sum.omdoc?sum?SumCollφiOMFOREIGNscala.xml.Node$@6040b6ef und iSaihttp://mathhub.info/smglom/numberfields/sum.omdoc?sum?SumInCollSiOMFOREIGNscala.xml.Node$@6040b6ef. Der erste spezifiziert den Summationsbereich durch eine Formel φφ über ii und der zweite gibt den Summationsbereich als eine Menge SS an.


  • Urbild Show Notations en

    Sei f:ABhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfAB eine Funktion, AAhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efA und BBhttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efB, dann nennen wir

    • (fundefeq[fname.cvar.2]f(A))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfname.cvar.2http://mathhub.info/smglom/sets/image.omdoc?image?imageoffOMFOREIGNscala.xml.Node$@6040b6ef das Bild von AOMFOREIGNscala.xml.Node$@6040b6ef unter ff,

    • (fundefeq[f]𝐈𝐦(f))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfhttp://mathhub.info/smglom/sets/image.omdoc?image?imagef das Bild von ff, und

    • (fundefeq[fname.cvar.6]f-1(B))http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqfname.cvar.6http://mathhub.info/smglom/sets/image.omdoc?image?pre-imagefOMFOREIGNscala.xml.Node$@6040b6ef das Urbild von BOMFOREIGNscala.xml.Node$@6040b6ef unter ff.


  • Vektor en

    Der nn-dimenionale Cartesische Raum Anhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?nCartSpaceAn über einer Menge AA ist definiert als (bsetst[an](a1,,an))http://mathhub.info/smglom/sets/set.omdoc?set?bsetstanhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlian. Wir nennen ein element ((a1,,an))(An)http://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlianhttp://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?nCartSpaceAn einen Vektor.


  • Vektorprodukt Show Notations en

    Sei VV ein nn-dimensionaler Vektorraum und {(v1,,vn)}http://mathhub.info/smglom/sets/set.omdoc?set?sethttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlivn eine Basis VV, dann definieren wir das Vektorprodukt der Vektoren w1,...,w(n-1)http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?nseqliwhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn durch

    (nfundefeqli[w]1)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?nfundefeqliw

  • verallgemeinerte Cullen-Primzahl en

    Eine verallgemeinerte Cullen-Zahl ist eine natürliche Zahl der Form

    (nbn)+1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationbn

    mit (n+2)>bhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnb.

    Wenn eine Primzahl in dieser Form dargestellt werden kann, nennt man sie eine verallgemeinerte Cullen-Primzahl.


  • verallgemeinerte Cullen-Zahl en

    Eine verallgemeinerte Cullen-Zahl ist eine natürliche Zahl der Form

    (nbn)+1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationbn

    mit (n+2)>bhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnb.

    Wenn eine Primzahl in dieser Form dargestellt werden kann, nennt man sie eine verallgemeinerte Cullen-Primzahl.


  • Verallgemeinerte harmonische Reihen en

    Verallgemeinerte harmonische Reihen sind Reihen der Form

    (infinite-sum0[n]1((an)+b))http://mathhub.info/smglom/calculus/infinitesum.omdoc?infinitesum?infinite-sumnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationanb

    mit reellen Zahlen a0http://mathhub.info/smglom/mv/equal.omdoc?equal?notequala und bb.


  • verallgemeinerte Woodall-Primzahl

    Eine verallgemeinerte Woodall-Zahl ist eine natürliche Zahl der Form

    (nbn)-1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationbn

    mit (n+2)>bhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnb.

    Wenn eine Primzahl in dieser Form dargestellt werden kann, nennt man sie eine verallgemeinerte Woodall-Primzahl.


  • verallgemeinerte Woodall-Zahl en

    Eine verallgemeinerte Woodall-Zahl ist eine natürliche Zahl der Form

    (nbn)-1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationbn

    mit (n+2)>bhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionnb.

    Wenn eine Primzahl in dieser Form dargestellt werden kann, nennt man sie eine verallgemeinerte Woodall-Primzahl.


  • Vereinigung Show Notations tr ro ru

    Ist SS eine Familie von Mengen, so ist die Vereinigung iISihttp://mathhub.info/smglom/sets/union.omdoc?union?munionCollectionIiOMFOREIGNscala.xml.Node$@6040b6ef über SS gegeben als (bsetst[x]x)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstxx.


  • Vereinigung Show Notations tr ro ru

    Sind AA und BB Mengen, so heißt (bsetst[x]x)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstxx die Vereinigung ABhttp://mathhub.info/smglom/sets/union.omdoc?union?unionAB von AA und BB.


  • Verkettung Show Notations en

    Die Verkettung zweier Relationen R(A×B)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetRhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAB und S(B×C)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetShttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsBC ist definiert as (fundefeq[SR]SR)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqSRhttp://mathhub.info/smglom/sets/relation-composition.omdoc?relation-composition?compositionSR


  • vielfache Harshad-Zahl en

    Eine vielfache Harshad-Zahl ist eine Harshad-Zahl die nach Division durch die Summe ihrer Ziffern wieder eine Harshad-Zahl ist.


  • Vielfachen en

    Sei Rhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureR ein Ring und a,bRhttp://mathhub.info/smglom/sets/set.omdoc?set?minsetabR mit a0http://mathhub.info/smglom/mv/equal.omdoc?equal?notequala, dann nennen wir aa einen linken Teiler von bb (und sagen aa teilt bb links oder bb ist link teilbar durch aa) und bb ein linkes Vielfaches von aa, wenn es ein nRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnR gibt, für die (multiplication)=bhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationb ist.

    Analog nennen wir aa einen rechten Teiler von bb (und sagen aa teilt bb rechts oder bb ist rechts teilbar durch aa) und bb ein rechtes Vielfaches von aa, wenn es ein nRhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnR gibt, für die (na)bhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnab ist.

    Ist RR kommutativ, so fallen rechte und linke Teiler zusammen, und wir sprechen einfach von Teiler, teilbar, und Vielfachen. Wir schreiben dann a|bhttp://mathhub.info/smglom/algebra/ring-divisor.omdoc?ring-divisor?divisorab.

    Wir nennen 1, 1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtraction, nn und nhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionn triviale divisoren von nn. Ein Teiler von nn, der nicht trivial ist, heißt echter Teiler von nn.


  • vierzehnte Smarandache-Konstante Show Notations

    Die vierzehnte Smarandache-Konstante fourteenth-smarandache-constanthttp://mathhub.info/smglom/smglom/smarandacheconstant14.omdoc?smarandacheconstant14?fourteenth-smarandache-constant ist definiert als

    s14(α)http://mathhub.info/smglom/smglom/smarandacheconstant14.omdoc?smarandacheconstant14?fourteenth-smarandache-constantα

    Dabei ist S(n)http://mathhub.info/smglom/smglom/smarandachefunction.omdoc?smarandachefunction?smarandache-functionn die Smarandache-Funktion.

    Die Summe konvergiert für alle reellen Zahlen α>1http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanα.


  • vollkommene Zahl en

    Eine natürliche Zahl nn wird vollkommene Zahl (auch perfekte Zahl) genannt, wenn sie gleich der Summe aller ihrer (positiven) echten Teiler ist.


  • vollständig en

    Ein metrischer Raum M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd heißt vollständig (oder vollständiger Raum), wenn in ihm jede Cauchyfolge konvergent ist.


  • vollständig en

    Wenn in einem Graph GG alle Ecken paarweise adjazent sind, dann nennen wir GG vollständig. A vollständiger Graph auf nn Ecken wird mit Knhttp://mathhub.info/smglom/graphs/graphcomplete.omdoc?graphcomplete?completegraphn bezeichnet.


  • vollständiger Graph Show Notations en

    Wenn in einem Graph GG alle Ecken paarweise adjazent sind, dann nennen wir GG vollständig. A vollständiger Graph auf nn Ecken wird mit Knhttp://mathhub.info/smglom/graphs/graphcomplete.omdoc?graphcomplete?completegraphn bezeichnet.


  • vollständiger Raum en

    Ein metrischer Raum M,dhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureMd heißt vollständig (oder vollständiger Raum), wenn in ihm jede Cauchyfolge konvergent ist.


  • Wagstaff-Primzahl en

    Eine Wagstaff-Primzahl ist eine Primzahl pp der Form

    p=(((2q)+1)3)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationq

    wobei qq auch eine Primzahl ist.


  • Wertebereich Show Notations en

    Eine Relation f(X×Y)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetfhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsXY, heißt partielle Funktion mit Argumentbereich XX (schreibe 𝐝𝐨𝐦(f)http://mathhub.info/smglom/sets/functions.omdoc?functions?domainf) und Wertebereich YY (schreibe 𝐜𝐨𝐝𝐨𝐦(f)http://mathhub.info/smglom/sets/functions.omdoc?functions?codomainf), wenn es für jedes xXhttp://mathhub.info/smglom/sets/set.omdoc?set?insetxX höchstens ein yYhttp://mathhub.info/smglom/sets/set.omdoc?set?insetyY gibt mit (x,y)fhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairxyf.

    Wir schreiben f:XY;xyhttp://mathhub.info/smglom/sets/functions.omdoc?functions?partfunsuchthatfXYxy und (f)=yhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalfy wenn (x,y)fhttp://mathhub.info/smglom/sets/set.omdoc?set?insethttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairxyf.


  • Wieferich-Primzahl en

    Eine Wieferich-Primzahl ist eine Primzahl pp mit der Eigenschaft, dass (2(p-1))-1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionp durch p2http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationp teilbar ist, also

    (2(p-1))1mod(p2)http://mathhub.info/smglom/numberfields/congruence.omdoc?congruence?congruent-modulohttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionphttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationp

  • Wilson-Primzahl en

    Eine Wilson-Primzahl ist eine Primzahl pp, so dass p2http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationp den Ausdruck ((p-1)!)+1http://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/factorial.omdoc?factorial?factorialhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionp teilt.


  • Wilson-Quotient en

    Für ganze integer mm wird der Wilson-Quotient W(m)http://mathhub.info/smglom/smglom/wilsonquotient.omdoc?wilsonquotient?Wilson-Quotientm wie folgt definiert

    (W(m))=((((m-1)!)+1)m)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/smglom/wilsonquotient.omdoc?wilsonquotient?Wilson-Quotientmhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/factorial.omdoc?factorial?factorialhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionmm

  • Wilson-Zahl en

    Eine Wilson-Zahl ist eine ganze Zahl nn, für die (W(n))0modnhttp://mathhub.info/smglom/numberfields/congruence.omdoc?congruence?congruent-modulohttp://mathhub.info/smglom/smglom/wilsonquotient.omdoc?wilsonquotient?Wilson-Quotientnn gilt. Dabei ist W(n)http://mathhub.info/smglom/smglom/wilsonquotient.omdoc?wilsonquotient?Wilson-Quotientn der Wilson-Quotient (((n-1)!)+1)nhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/factorial.omdoc?factorial?factorialhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnn.


  • Wolstenholme-Primzahl en

  • Woodall-Primzahlen

    Eine Woodall-Zahl ist eine natürliche Zahl der Form

    (Wn)=((n2n)-1)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/woodallnumber.omdoc?woodallnumber?Woodall-numbernhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn

    mit einer natürlichen Zahl nn. Woodall-Zahlen werden manchmal auch als Cullen-Zahlen der zweiten Art bezeichnet.

    Woodall-Zahlen die Primzahlen sind werden als Woodall-Primzahlen bezeichnet.


  • Woodall-Zahl Show Notations en

    Eine Woodall-Zahl ist eine natürliche Zahl der Form

    (Wn)=((n2n)-1)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/woodallnumber.omdoc?woodallnumber?Woodall-numbernhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn

    mit einer natürlichen Zahl nn. Woodall-Zahlen werden manchmal auch als Cullen-Zahlen der zweiten Art bezeichnet.

    Woodall-Zahlen die Primzahlen sind werden als Woodall-Primzahlen bezeichnet.


  • Woodall-Zahlen Show Notations en

    Eine Woodall-Zahl ist eine natürliche Zahl der Form

    (Wn)=((n2n)-1)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/woodallnumber.omdoc?woodallnumber?Woodall-numbernhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?subtractionnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationn

    mit einer natürlichen Zahl nn. Woodall-Zahlen werden manchmal auch als Cullen-Zahlen der zweiten Art bezeichnet.

    Woodall-Zahlen die Primzahlen sind werden als Woodall-Primzahlen bezeichnet.


  • Zahlentheorie en

    Die Zahlentheorie ist ein Teilgebiet der Mathematik, das sich mit den Eigenschaften der ganzen Zahlen beschäftigt.


  • Zehnerlogarithmus Show Notations en

    Der dekadische Logarithmus (oder Zehnerlogarithmus) ist der Logarithmus zur Basis 10.


  • zehnten Smarandache-Konstanten Show Notations

    Die zehnten Smarandache-Konstanten http://mathhub.info/smglom/smglom/smarandacheconstant10.omdoc?smarandacheconstant10?tenth-smarandache-constant sind definiert als

    s10(α)http://mathhub.info/smglom/smglom/smarandacheconstant10.omdoc?smarandacheconstant10?smarandacheconsttenα

    Dabei ist S(n)http://mathhub.info/smglom/smglom/smarandachefunction.omdoc?smarandachefunction?smarandache-functionn die Smarandache-Funktion.

    Die Summe konvergiert für alle reellen Zahlen α>1http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?morethanα.


  • Zentral-Polygonal-Zahl en

    Eine Zentral-Polygonal-Zahl ist die größte Zahl pp von Teilen, in die ein Kreis (oder eine Ebene) durch nn gerade Schnitte zerlegt werden kann.

    (multi-relation-expressionpequal(𝒞2n)+(𝒞1n)+(𝒞0n)equal((n2)+n+2)2)http://mathhub.info/smglom/sets/multirel.omdoc?multirel?multi-relation-expressionphttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficientnhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficientnhttp://mathhub.info/smglom/smglom/binomialcoefficient.omdoc?binomialcoefficient?binomial-coefficientnhttp://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?divisionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?additionhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationnn

  • Zentrale Delannoy-Zahlen Show Notations en

    Zentrale Delannoy-Zahlen (D(n))=(D(n,n))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/numbers/centraldelannoynumber.omdoc?centraldelannoynumber?central-Delannoy-numbernhttp://mathhub.info/smglom/numbers/delannoynumber.omdoc?delannoynumber?Delannoy-numbernn sind Delannoy-Zahlen für ein quadratisches nnhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationnn Gitter.

    Die ersten zentralen Delannoy-Zahlen sind:

    1,3,13,63,321,1683,8989,48639,265729,...http://mathhub.info/smglom/calculus/sequences.omdoc?sequences?infseq

  • zirkulare Primzahl en

    Eine Primzahl ist eine zirkulare Primzahl, wenn alle Zahlen, die durch zyklische Vertauschung ihrer Ziffern entstehen, auch Primzahlen sind.

    Zum Beispiel ist 197 eine zirkulare Primzahl, weil 971 und 719 auch Primzahlen sind.


  • zusammengesetzte Zahl en

    Eine zusammengesetzte Zahl ist eine natürlicheZahl, die mindestens drei verschiedene Teiler besitzt.


  • zusammenhängender en

    Ein nicht leerer Graph GG wird zusammenhängenderenannt, wenn je zwei seiner Ecken durch einen Weg verbunden sind. Wenn für eine Teilmenge UU der Eckenmenge VGhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationVG der induzierte Teilgraph GUhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationGU zusammenhängend ist, so nennen wir UU zusammenhängend in GG. Als Verneinung ziehen wir ‘unzusammenhängend’ der Formulierung ‘nicht zusammenhängend’ vor.


  • zweite Ableitung en

    Wir nennen ff nn mal differeinzierbar in aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM, falls dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa als Grenzwert existiert. Analog nennen wir ff nn mal differenzierbar auf MM, falls dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx existiert und auf MM total ist.

    Wir nennen ff unendlich differenzierbar oder glatt auf aMhttp://mathhub.info/smglom/sets/set.omdoc?set?insetaM oder MM, wenn dnfdxn|x=ahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa beziehungsweise dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx für alle nhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number existieren.

    Für ein nhttp://mathhub.info/smglom/sets/set.omdoc?set?insetnhttp://mathhub.info/smglom/numberfields/naturalnumbers.omdoc?naturalnumbers?natural-number definieren wir die nnte Ableitung einer Funktion f:MNhttp://mathhub.info/smglom/sets/functions.omdoc?functions?funfMN durch

    (fundefeq[nfxa]dnfdxn|x=a)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqnfxahttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtatnfxa

    Die erste Ableitung d1dx1fhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtfx von ff ist Dxfhttp://mathhub.info/smglom/calculus/derivative.omdoc?derivative?derivativewrtfx, d2dx2fhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtfx ist die zweite Ableitung von ff, d3dx3fhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtfx die dritte Ableitung von ff, usw. In der Leibniz Notation wird die nnte Ableiguntsfunktion von ff als dndxnfhttp://mathhub.info/smglom/calculus/nderivative.omdoc?nderivative?nderivativewrtnfx geschrieben.


  • zweite Skewes-Zahl Show Notations en

    Die zweite Skewes-Zahl http://mathhub.info/smglom/smglom/skewesnumber.omdoc?skewesnumber?second-Skewes-number ist eine Obergrenze bis zu der nicht immer (π(n))(Li(n))http://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?lethanhttp://mathhub.info/smglom/primes/primenumber.omdoc?primenumber?NumberPrimeNumbernhttp://mathhub.info/smglom/smglom/logarithmicintegralbig.omdoc?logarithmicintegralbig?logarithmicintbign gilt, vorausgesetzt, die Riemann-Hypothese ist falsch.

    =(10(10(101000)))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/smglom/skewesnumber.omdoc?skewesnumber?second-Skewes-numberhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiationhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?exponentiation

  • zweite Smarandache-Konstante Show Notations en

  • zweite Tschebyschow-Funktion en

    Die zweite Tschebyschow-Funktion ψ(x)http://mathhub.info/smglom/smglom/chebyshevfunction.omdoc?chebyshevfunction?secchebyfuncx ist die Summe der Logarithmen der Primzahlen über alle Primzahlpotenzen bis xx.

    ψ(x)http://mathhub.info/smglom/smglom/chebyshevfunction.omdoc?chebyshevfunction?secchebyfuncx

    Dabei ist ΛΛ die Mangoldt-Funktion.


  • Zwischenprimzahl en

    Eine Zwischenprimzahl ist das arithmetische Mittel zweier aufeinanderfolgender ungerader Primzahlen.


  • Zykel en

    Ist G=(V,E)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalGhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE ein gerichteter Graph, so nennen wir

    • einen Pfad pp in GG zyklisch (auch einen Zykel oder eine Schleife), falls (start(p))=(end(p))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pstartphttp://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pendp.

    • einen Zykel (v0,,vn)http://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlivn einfach, wenn vivjhttp://mathhub.info/smglom/mv/equal.omdoc?equal?notequalOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6ef für alle i,j1nhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?mbetweeneeijn mit ijhttp://mathhub.info/smglom/mv/equal.omdoc?equal?notequalij.

    • GG azyklisch, wenn GG keinen Zykel enthält.


  • zyklisch en

    Ist G=(V,E)http://mathhub.info/smglom/mv/equal.omdoc?equal?equalGhttp://mathhub.info/smglom/mv/structure.omdoc?structure?structureVE ein gerichteter Graph, so nennen wir

    • einen Pfad pp in GG zyklisch (auch einen Zykel oder eine Schleife), falls (start(p))=(end(p))http://mathhub.info/smglom/mv/equal.omdoc?equal?equalhttp://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pstartphttp://mathhub.info/smglom/graphs/graph-path.omdoc?graph-path?pendp.

    • einen Zykel (v0,,vn)http://mathhub.info/smglom/sets/cartesian-product.omdoc?cartesian-product?tuplehttp://mathhub.info/smglom/mv/argseq.omdoc?argseq?naseqlivn einfach, wenn vivjhttp://mathhub.info/smglom/mv/equal.omdoc?equal?notequalOMFOREIGNscala.xml.Node$@6040b6efOMFOREIGNscala.xml.Node$@6040b6ef für alle i,j1nhttp://mathhub.info/smglom/numberfields/numbers-orders.omdoc?numbers-orders?mbetweeneeijn mit ijhttp://mathhub.info/smglom/mv/equal.omdoc?equal?notequalij.

    • GG azyklisch, wenn GG keinen Zykel enthält.


  • zyklische Zahl en

    Eine zyklische Zahl ist eine natürliche Zahl, deren zyklische Permutationen ihrer Ziffern aufeinanderfolgende Vielfache dieser Zahl sind.

    (1428571)=142857http://mathhub.info/smglom/mv/equal.omdoc?equal?equal
    (1428572)=285714http://mathhub.info/smglom/mv/equal.omdoc?equal?equal
    (1428573)=428571http://mathhub.info/smglom/mv/equal.omdoc?equal?equal
    (1428574)=571428http://mathhub.info/smglom/mv/equal.omdoc?equal?equal
    (1428575)=714285http://mathhub.info/smglom/mv/equal.omdoc?equal?equal
    (1428576)=857142http://mathhub.info/smglom/mv/equal.omdoc?equal?equal

  • ägyptischer Bruch en

    Ein ägyptischer Bruch ist die Summe von verschiedenen Stammbrüchen.


  • Äquivalenzklasse Show Notations en

    Seien SS eine Menge, RR eine Äquivalenzrelation auf SS und xShttp://mathhub.info/smglom/sets/set.omdoc?set?insetxS, dann nennen wir die Menge (fundefeq[xR][x]R)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqxRhttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?equivalence-classxR die Äquivalenzklasse von xx (unter RR), und die Menge (fundefeq[SR]S_R)http://mathhub.info/smglom/sets/fundefeq.omdoc?fundefeq?fundefeqSRhttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?quotient-spaceSR die Quotientenmenge von SS (unter RR).

    Wir nennen die Abbildung (projectionR):S(S_R);x([x]R)http://mathhub.info/smglom/sets/functions.omdoc?functions?funsuchthathttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?projectionRShttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?quotient-spaceSRxhttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?equivalence-classxR die Projektion von SS auf S_Rhttp://mathhub.info/smglom/sets/quotientspace.omdoc?quotientspace?quotient-spaceSR unter RR.


  • Äquivalenzrelation en

    Eine Relation R(A×A)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetRhttp://mathhub.info/smglom/sets/pair.omdoc?pair?pairsAA ist eine Äquivalenzrelation auf AA, wenn RR reflexiv, symmetrisch und transitiv ist.


  • Überdeckung en

    Eine Überdeckung einer Menge XX ist eine Familie CC von Mengen mit X(aAUa)http://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetXhttp://mathhub.info/smglom/sets/union.omdoc?union?munionCollectionAaOMFOREIGNscala.xml.Node$@6040b6ef. Eine Teilmenge CChttp://mathhub.info/smglom/sets/subsupset.omdoc?subsupset?subsetOMFOREIGNscala.xml.Node$@6040b6efC heißt Teilüberdeckung von XX, wenn sie XX immer noch überdeckt.


  • power set Show Notations de ro tr

    Let AA be a set, then the power set 𝒫(A)http://mathhub.info/smglom/sets/powerset.omdoc?powerset?powersetA of AA is (bsetst[S]S)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstSS.


  • set difference Show Notations de ro tr

    Let AA and BB be sets, then the set difference A\Bhttp://mathhub.info/smglom/sets/setdiff.omdoc?setdiff?set-differenceAB of AA and BB is (bsetst[x]x)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstxx.


  • equal Show Notations tr ro de

    Two sets List(\ttl)Ahttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationA and List(\ttl)Bhttp://mathhub.info/smglom/numberfields/arithmetics.omdoc?arithmetics?multiplicationB are equal (written ABhttp://mathhub.info/smglom/sets/set.omdoc?set?setequalAB), iff they have the same elements.


  • union Show Notations de tr ro

    Let II be a set and (bsetst[i]Si)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstiOMFOREIGNscala.xml.Node$@6040b6ef a family of sets, then the union iISihttp://mathhub.info/smglom/sets/union.omdoc?union?munionCollectionIiOMFOREIGNscala.xml.Node$@6040b6ef over the collection SS is (bsetst[x]x)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstxx.


  • union Show Notations de tr ro

    Let AA and BB be sets, then the union ABhttp://mathhub.info/smglom/sets/union.omdoc?union?unionAB of AA and BB is defined as (bsetst[x]x)http://mathhub.info/smglom/sets/set.omdoc?set?bsetstxx